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6.3
Proofs with Parallel Lines
For use with Exploration 6.3
Name _________________________________________________________ Date _________
Essential Question For which of the theorems involving parallel lines
and transversals is the converse true?
Work with a partner. Write the converse of each conditional statement. Draw a
diagram to represent the converse. Determine whether the converse is true. Justify
your conclusion.
a. Corresponding Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of
corresponding angles are congruent.
Converse
b. Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of
alternate interior angles are congruent.
Converse
c. Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of
alternate exterior angles are congruent.
Converse
1
4
2
3
6
7
8
5
1
4
2
3
6
7
8
5
1
4
2
3
6
7
8
5
1 EXPLORATION: Exploring Converses
193
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6.3
Proo
f
s with Parallel Lines (continued)
Name _________________________________________________________ Date __________
d. Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of
consecutive interior angles are supplementary.
Converse
Communicate Your Answer
2. For which of the theorems involving parallel lines and transversals is the
converse true?
3. In Exploration 1, explain how you would prove any of the theorems that
you found to be true.
1 EXPLORATION: Exploring Converses (continued)
1
4
2
3
6
7
8
5
194
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6.3
For use after Lesson 6.3
Name _________________________________________________________ Date _________
In your own words, write the meaning of each vocabulary term.
converse
parallel lines
transversal
corresponding angles
congruent
alternate interior angles
alternate exterior angles
consecutive interior angles
Theorems
Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles
are congruent, then the lines are parallel.
Notes: jk
j
k
6
2
Copyright © Big Ideas Learning, LLC
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6.3
For use after Lesson 6.3
Name _________________________________________________________ Date _________
In your own words, write the meaning of each vocabulary term.
converse
parallel lines
transversal
corresponding angles
congruent
alternate interior angles
alternate exterior angles
consecutive interior angles
Theorems
Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles
are congruent, then the lines are parallel.
Notes: jk
j
k
6
2
195
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6.3
Notetaking with Vocabulary
For use after Lesson 6.3
Name _________________________________________________________ Date _________
In your own words, write the meaning of each vocabulary term.
converse
parallel lines
transversal
corresponding angles
congruent
alternate interior angles
alternate exterior angles
consecutive interior angles
Theorems
Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles
are congruent, then the lines are parallel.
Notes: jk
j
k
6
2
196
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Notetakin
g
with Vocabular
y
(continued)
6.3
Name _________________________________________________________ Date __________
Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles
are congruent, then the lines are parallel.
Notes:
Alternate Exterior Angles Converse
If two lines are cut by a transversal so the alternate exterior angles are
congruent, then the lines are parallel.
Notes:
Consecutive Interior Angles Converse
If two lines are cut by a transversal so the consecutive interior angles are
supplementary, then the lines are parallel.
Notes:
If two lines are parallel to the same line, th
Notes:
k
j
5
4
k
j
1
8
k
j
3
5
p
r
q
jk
j
k
If
3 and 5∠∠
are
supplementary, then
.jk
If and , then
.
pq qr
pr
196
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Notetakin
g
with Vocabular
y
(continued)
6.3
Name _________________________________________________________ Date __________
Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles
are congruent, then the lines are parallel.
Notes:
Alternate Exterior Angles Converse
If two lines are cut by a transversal so the alternate exterior angles are
congruent, then the lines are parallel.
Notes:
Consecutive Interior Angles Converse
If two lines are cut by a transversal so the consecutive interior angles are
supplementary, then the lines are parallel.
Notes:
Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other.
Notes:
k
j
5
4
k
j
1
8
k
j
3
5
p
r
q
jk
j
k
If
3 and 5∠∠
are
supplementary, then
.jk
If and , then
.
pq qr
pr
196
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Notetakin
g
with Vocabular
y
(continued)
6.3
Name _________________________________________________________ Date __________
Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles
are congruent, then the lines are parallel.
Notes:
Alternate Exterior Angles Converse
If two lines are cut by a transversal so the alternate exterior angles are
congruent, then the lines are parallel.
Notes:
Consecutive Interior Angles Converse
If two lines are cut by a transversal so the consecutive interior angles are
supplementary, then the lines are parallel.
Notes:
Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other.
Notes:
k
j
5
4
k
j
1
8
k
j
3
5
p
r
q
jk
j
k
If
3 and 5∠∠
are
supplementary, then
.jk
If and , then
.
pq qr
pr
196
Copyright © Big Ideas Learning, LLC
All rights reserved.
Notetakin
g
with Vocabular
y
(continued)
6.3
Name _________________________________________________________ Date __________
Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles
are congruent, then the lines are parallel.
Notes:
Alternate Exterior Angles Converse
If two lines are cut by a transversal so the alternate exterior angles are
congruent, then the lines are parallel.
Notes:
Consecutive Interior Angles Converse
If two lines are cut by a transversal so the consecutive interior angles are
supplementary, then the lines are parallel.
Notes:
Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other.
Notes:
k
j
5
4
k
j
1
8
k
j
3
5
p
r
q
jk
j
k
If
3 and 5∠∠
are
supplementary, then
.jk
If and , then
.
pq qr
pr
195
Copyright © Big Ideas Learning, LLC
All rights reserved.
6.3
Notetaking with Vocabulary
For use after Lesson 6.3
Name _________________________________________________________ Date _________
In your own words, write the meaning of each vocabulary term.
converse
parallel lines
transversal
corresponding angles
congruent
alternate interior angles
alternate exterior angles
consecutive interior angles
Theorems
Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles
are congruent, then the lines are parallel.
Notes:
jk
j
k
6
2
Practice
195
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6.3
Name _________________________________________________________ Date __________
Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles
are congruent, then the lines are parallel.
k
p
r
q
3 and 5
are
.jk
.
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6.3
Name _________________________________________________________ Date __________
Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles
are congruent, then the lines are parallel.
k
p
r
q
3 and 5
are
.jk
.
Worked-Out Examples
Example #1
Example #2
Integrated Mathematics I 461
Worked-Out Solutions
Chapter 10
7. Lines m and n are parallel when the marked consecutive
interior angles are supplementary.
x° + 2x° = 180°
3x = 180
3x
—
3
=
180
—
3
x = 60
8. Lines m and n are parallel when the marked alternate interior
angles are congruent.
3x° = (2x + 20)°
x = 20
9. Let A and B be two points on line m. Draw
AP and
construct an angle ∠ 1 on n at P so that ∠ PAB and ∠ 1 are
corresponding angles.
P
AB
1
m
n
10. Let A and B be two points on line m. Draw
AP and
construct an angle ∠ 1 on n at P so that ∠ PAB and ∠ 1 are
corresponding angles.
P
A
B
n
m
1
12. Given ∠ 3 and ∠ 5 are supplementary.
k
j
32
5
Prove j
k
STATEMENTS REASONS
1. ∠ 3 and ∠ 5 are
supplementary.
1. Given
2. ∠ 2 and ∠ 3 are
supplementary.
2. Linear Pair Postulate
3. m∠ 3 + m∠ 5 = 180°,
m∠ 2 + m∠ 3 = 180°
3. De nition of
supplementary angles
4. m∠ 3 + m∠ 5 =
m∠ 2 + m∠ 3
4.
Transitive Property of
Equality
5. m∠ 2 = m∠ 5
5.
Subtraction Property of
Equality
6. ∠ 2 ≅ ∠ 5
6. De nition of congruent
angles
7. j
k
7. Corresponding Angles
Converse
13. yes; Alternate Interior Angles Converse
= y, and
∠ 2
Copyright © Big Ideas Learning, LLC Integrated Mathematics I 461
All rights reserved. Worked-Out Solutions
Chapter 10
7. Lines m and n are parallel when the marked consecutive
interior angles are supplementary.
x° + 2x° = 180°
3x = 180
3x
—
3
=
180
—
3
x = 60
8. Lines m and n are parallel when the marked alternate interior
angles are congruent.
3x° = (2x + 20)°
x = 20
9. Let A and B be two points on line m. Draw
AP and
construct an angle ∠ 1 on n at P so that ∠ PAB and ∠ 1 are
corresponding angles.
P
AB
1
m
n
10. Let A and B be two points on line m. Draw
AP and
construct an angle ∠ 1 on n at P so that ∠ PAB and ∠ 1 are
corresponding angles.
P
A
B
n
m
1
11. Given ∠ 1 ≅ ∠ 8
k
j
1
2
8
Prove j
k
STATEMENTS REASONS
1. ∠ 1 ≅ ∠ 8
1. Given
2. ∠ 1 ≅ ∠ 2
2. Vertical Angles Congruence
Theorem
3. ∠ 8 ≅ ∠ 2
3. Transitive Property of Congruence
4. j
k
4. Corresponding Angles Converse
13. yes; Alternate Interior Angles Converse
14. yes; Alternate Exterior Angles Converse
15. no
16. yes; Corresponding Angles Converse
17. no
18. yes; Alternate Exterior Angles Converse
19. This diagram shows that vertical angles are always
congruent. Lines a and b are not parallel unless x = y, and
you cannot assume that they are equal.
20. It would be true that a
b if you knew that ∠ 1 and ∠2
were supplementary, but you cannot assume that they are
supplementary unless it is stated or the diagram is marked as
such. You can say that ∠ 1 and ∠ 2 are consecutive interior
angles.
21. yes; m∠ DEB = 180° − 123° = 57° by the Linear Pair
Postulate. So, by de nition, a pair of corresponding
angles are congruent, which means that
AC
DF by the
Corresponding Angles Converse.
22. yes; m∠ BEF = 180° − 37° = 143° by the Linear Pair
Postulate. So, by de nition, a pair of corresponding
angles are congruent, which means that
AC
DF by the
Corresponding Angles Converse.
23. cannot be determined; The marked angles are vertical angles.
You do not know anything about the angles formed by the
intersection of
DF and
BE .
Practice (continued)
196
Lines m and n are parallel when the marked consecutive
interior angles are supplementary.
180° = 150° + (3x − 15)°
180 = 135 + 3x
45 = 3x
45
—
3
=
3x
—
3
x = 15
Lines m and n are parallel when the marked alternate exterior
angles are congruent.
x° = (180 − x)°
2x = 180
2x
—
2
=
180
—
2
x = 90
n
m
150°
(3x − 15)°
nm
(180 − x)°
x°
Find the value of x that makes m || n.
Explain your reasoning.
Find the value of x that makes m || n.
Explain your reasoning.
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6.3
Name _________________________________________________________ Date _________
Extra Practice
In Exercises 1 and 2, find the value of x that makes
mn.
Explain your reasoning.
1. 2.
In Exercises 3–6, decide whether there is enough information to prove that
mn.
If so, state the theorem you would use.
3. 4.
5. 6.
m
n
(8x + 55)°
95°
m
n
r
s
m
r
n
m
s
r
n
130°
(200 − 2x)°
m
n
m
r
n
Practice A
(continued)Practice
197
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10.3
Practice B
Name _________________________________________________________ Date __________
In Exercises 1 and 2, find the value of x that makes st. Explain your reasoning.
1. 2.
In Exercises 3 and 4, decide whether there is enough information to prove that
pq.
If so, state the theorem you would use.
3. 4.
5. The map of the United States shows the lines of latitude and
longitude. The lines of latitude run horizontally and the lines
of longitude run vertically.
a. Are the lines of latitude parallel? Explain.
b. Are the lines of longitude parallel? Explain.
6. Use the diagram to answer the following. 7. Given: 1 2 and 2 3∠ ≅∠ ∠ ≅∠
Prove:
14∠ ≅∠
a. Find the values of x, y, and z that makes
p
q and .qr Explain your reasoning.
b. Is ?
p
r Explain your reasoning.
s
r
t
(7x − 20)°
(4x + 16)°
2(x + 15)°
st
r
(3x + 20)°
r
qp
r
q
p
(180
− x)°
x
°
a
b
c
d
r
s
qp
3(x
−
1)°
(4x
−
30)° (6y)°
6(z + 8)°
1
2
4
3
b
c
d
a
Practice B
198
