Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage1
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage2
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage3
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage4
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage5
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage6
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage7
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage8
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage9
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
eSolutionsManual-PoweredbyCogneroPage10
5-6 Proving Lines Parallel
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1.
SOLUTION:
and
are corresponding angles of lines j and k.
Since ,
j
|| k by the Converse of Corresponding Angles
Postulate.
2.
SOLUTION:
and
are alternate interior angles of lines j and
k. Since ,
j
|| k by the Converse of Alternate Interior Angles
Theorem.
3.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
4.m 6 + m 8 = 180
SOLUTION:
and
are consecutive interior angles of lines
and m. Since , || m by the
Converse of Consecutive Interior Angles Theorem.
5.
SHORT RESPONSE
Find x so that m || n. Show
your work.
SOLUTION:
angle and
angle
are alternate
exterior angles of lines m and n. Since
m
|| n, by the Converse of
Alternate Exterior Angles Theorem.
Solve for x.
6.
PROOF
Copy and complete the proof of Theorem
5.18.
Given:
Prove:
Proof:
SOLUTION:
7.
RECREATION
Is it possible to prove that the
backrest and footrest of the lounging beach chair are
parallel? If so, explain how. If not, explain why not.
SOLUTION:
Sample answer: Yes; since the alternate exterior
angles are congruent, the backrest and footrest are
parallel.
Given the following information, determine
which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
8.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
9.
SOLUTION:
and
are alternate exterior angles of lines
and m. Since ,
|| m by the Converse of
Alternate Exterior Angles Theorem.
10.
SOLUTION:
and
are alternate interior angles of lines r and
s. Since ,
r
|| s by the Converse of Alternate Interior Angles
Theorem.
11.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
12.
SOLUTION:
and
are consecutive interior angles of lines r
and s. Since ,
r
|| s by the Converse of Consecutive Interior Angles
Theorem.
13.
SOLUTION:
and
are alternate interior angles of lines u
and v. Since ,
u
|| v by the Converse of Alternate Interior Angles
Theorem.
14.
SOLUTION:
No lines can be proven parallel.
15.
SOLUTION:
and
are corresponding angles of lines r and s.
Since ,
r
|| s by the Converse of Corresponding Angles
Postulate.
Find x so that m || n. Identify the postulate or
theorem you used.
16.
SOLUTION:
By the Alternate Exterior Angles Converse, if 3x –
14 = 2x + 25, then m || n.
Solve for x.
17.
SOLUTION:
By the Converse of Corresponding Angles Postulate,
if 5x – 20 = 90, then m || n.
Solve for x.
18.
SOLUTION:
By the Alternate Interior Angles Converse, if 21 + 2x
= x + 84, then m || n.
Solve for x.
19.
SOLUTION:
By the Consecutive Interior Angles Converse, if 7x –
2 + 10 – 3x = 180, then m || n.
Solve for x.
20.
SOLUTION:
Use the Vertical Angle Theorem followed by
Consecutive Interior Angles Converse to find x.
Then by
Consecutive Interior Angles Converse, if 3x
+ 2x + 45 = 180, then m || n
.
Solve for x
.
21.
SOLUTION:
By the Alternate Exterior Angles Converse, if 6x
–
144 = 2x, then m || n
.
Solve for x.
22.
CCSS SENSE-MAKING
Wooden picture frames
are often constructed using a miter box or miter saw.
These tools allow you to cut at an angle of a given
size. If each of the four pieces of framing material is
cutata45°angle,willthesidesoftheframebe
parallel? Explain your reasoning.
SOLUTION:
Yes; when two pieces are put together, they form a
90°angle.Twolinesthatareperpendiculartothe
same line are parallel.
23.
PROOF
Copy and complete the proof of Theorem
5.19.
Given:
∠
1 and
∠
2 are supplementary.
Prove:
SOLUTION:
24.
CRAFTS
Jacqui is making a stained glass piece.
Shecutsthetopandbottompiecesata30°angle.If
the corners are right angles, explain how Jacqui
knows that each pair of opposite sides are parallel.
SOLUTION:
Since the corners are right angles, each pair of
opposite sides is perpendicular to the same line.
Therefore, each pair of opposite sides is parallel.
PROOF Write a two-column proof for each of
the following.
25.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Corr. postulate)
3.
(Trans.Prop.)
4.
(Ifalternate are
, then lines are
.)
26.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
2
3 (Given)
2. 2 and 4 are supplementary. (Cons. Int. s)
3. m
∠
2 + m
∠
4 = 180 (Def. of suppl.
∠
s)
4. m
∠
3+
m
∠
4 = 180 (Substitution)
5.
∠
3 and
∠
4 are supplementary. (Def. of suppl.
∠
s)
6.
(Ifcons.int.
s
are suppl., then lines
are .)
27.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1.
, (Given)
2.
(Def.of
)
3.
(Substitution)
4. and
aresupplementary.(Def.of
suppl. )
5.
(Ifconsec.int.
are suppl., then lines
are .)
28.
Given:

Prove:
SOLUTION:
Proof:
Statements (Reasons)
1. ,
(Given)
2.
(Ifalt.int.
are , then lines are .)
3.
(PerpendicularTransversalTheorem)
29.
MAILBOXES
Mail slots are used to make the
organization and distribution of mail easier. In the
mail slots shown, each slot is perpendicular to each
of the sides. Explain why you can conclude that the
slots are parallel.
SOLUTION:
The Converse of the Perpendicular Transversal
Theorem states that two coplanar lines perpendicular
to the same line are parallel. Since the slots are
perpendicular to each of the sides, the slots are
parallel. Since any pair of slots is perpendicular the
sides, they are also parallel.
30.
PROOF
Write a paragraph proof of Theorem 5.21.
SOLUTION:
Given:
Prove:
Proof:
Since
and
, the measures of angle 1 and
angle 2 are 90. Since
and havethesame
measure, they are congruent. By the converse of
Corresponding Angles Postulate, .
31.
PROOF
Write a two-column proof of Theorem
5.20.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1.
(Given)
2.
(Vertical
s
are )
3.
(TransitiveProp.)
4.
(Ifcorr are
, then lines are .)
32.
CCSS REASONING
Based upon the information
given in the photo of the staircase, what is the
relationship between each step? Explain your
answer.
SOLUTION:
Each step is parallel to each other because the
corresponding angles are congruent.
Determine whether lines r and s are parallel.
Justify your answer.
33.
SOLUTION:
r
||
s
;
Sample answer: The corresponding angles are
congruent. Since the measures of angles are equal,
the lines are parallel.
34.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
35.
SOLUTION:
r
|| s; Sample answer: The alternate exterior angles
are congruent. Since the measures of angles are
equal, the lines are parallel.
36.
MULTIPLE REPRESENTATIONS
In this
problem, you will explore the shortest distance
between two parallel lines.
a. GEOMETRIC
Draw three sets of parallel lines
k and , s and t, and x and y. For each set, draw the
shortest segment
andlabelpoints
A and D as
shown below.
b.
TABULAR
Copy the table below, measure
ABC and BCD, and complete the table.
c. VERBAL
Make a conjecture about the angle the
shortest segment forms with both parallel lines.
SOLUTION:
a.
b.
c.
Sample answer: The angle that the segment forms
with the parallel lines will always measure 90.
37.
ERROR ANALYSIS
Sumi and Daniela are
determining which lines are parallel in the figure at
the right. Sumi says that since 1
2, Danieladisagreesandsaysthat
since 1
2,
Iseitherofthem
correct? Explain.
SOLUTION:
Daniela;
and are alternate interior angles for
and , so if alternate interior angles are
congruent, then the lines are parallel.
38.
CCSS REASONING
Is Theorem 5.21 still true if
the two lines are not coplanar? Draw a figure to
justify your answer.
.
SOLUTION:
No; sample answer: In the figure shown,
and but
39.
CHALLENGE
Use the figure to prove that two
lines parallel to a third line are parallel to each other.
SOLUTION:
Given
: a || b and b || c
Prove
: a || c
Proof:
Statements (Reasons)
1. a || b and b || c (Given)
2.
∠
1
∠
3 (Alternate Interior
∠
's Theorem)
3.
∠
3
∠
2 (Vertical.
∠
's are )
4.
∠
2
∠
4 (Alternate Interior.
∠
's Theorem)
5.
∠
1
∠
4 (Trans. Prop.)
6. a || c (Alternate Interior.
∠
's Converse Theorem)
40.
OPEN ENDED
Draw a triangle ABC.
a.
Construct the line parallel to
throughpoint
A.
b.
Use measurement to justify that the line you
constructed is parallel to .
c.
Use mathematics to justify this construction.
SOLUTION:
a.
b.
Sample answer: Using a straightedge, the lines are
equidistant. So they are parallel.
c.
Sample answer:
isatransversalfor and
.
was copied to construct . So,
. and are
corresponding angles, so by the converse of
corresponding angles postulate,
41.
CHALLENGE
Refer to the figure at the right.
a.
If m 1 + m 2 = 180, prove that a || c.
b.
Given that a || c, if m 1 + m 3 = 180, prove that
t
⊥
c.
SOLUTION:
a
. We know that . Since and
are linear pairs, . By
substitution, . By
subtracting from both sides we get
. , by the definition of
congruent angles. Therefore,
sincethe
corresponding angles are congruent.
b
. We know that
and
. Since
and
are corresponding angles, they are
congruent and their measures are equal. By
substitution, . By dividing both
sides by 2, we get . Therefore, since
they form a right angle.
42.
WRITING IN MATH
Summarize the five methods
used in this lesson to prove that two lines are parallel.
SOLUTION:
Sample answer: Use a pair of alternate exterior
angles that are
congruent and cut by transversal; show that a pair of
consecutive interior angles are supplementary; show
that alternate interior angles are congruent; show two
coplanar lines are perpendicular to same line; show
corresponding angles are congruent.
43.
WRITING IN MATH
Can a pair of angles be
supplementary and congruent?
Explain your reasoning.
SOLUTION:
Yes; sample answer: A pair of angles can be both
supplementary and congruent if the measure of both
angles is 90, since the sum of the angle measures
would be 180.
44.
Which of the following facts would be sufficient to
prove that line d is parallel to
A
B
C
D
SOLUTION:
If line d is parallel to the line through then with
the transversals of , the alternate interior angles
must be congruent. Thus .
Thus B is the correct choice.
45.
ALGEBRA
The expression
is
equivalent to
F
13
G
H
J
SOLUTION:
So, the correct option is G.
46.
What is the approximate surface area of the figure?
A
101.3 in
2
B
108 in
2
C
202.5 in
2
D
216 in
2
SOLUTION:
The formula for finding the surface area of a prism is
.
S =
totalsurfacearea,
h = height of a solid, B =
area of the base, P = perimeter of the base
Since the base of the prism is a rectangle, the
perimeter P of the base is
or27inches.
The area of the base B is
or40.5squareinches.Theheightis5
cinches.
The surface area of the prism is 216 square inches.
So, the correct option is D.
47.
SAT/ACT
If x
2
= 25 and y
2
= 9, what is the greatest
possible value of (x
–
y)
2
?
F
4
G
16
H
58
J
64
K
70
SOLUTION:
First solve for x and y.
Next, find the greatest possible value. Substituting -
5 and 3 for x and y, respectively, leads to the
greatest positive number. Another solution is to
substitute 5 and -3 for x and y.
The correct choice is H.
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5-6 Proving Lines Parallel