Section 3.4 Proofs with Perpendicular Lines 147
Proofs with Perpendicular Lines
3.4
Writing Conjectures
Work with a partner. Fold a piece of paper
in half twice. Label points on the two creases,
as shown.
a. Write a conjecture about
—
AB and
—
CD .
Justify your conjecture.
b. Write a conjecture about
—
AO and
—
OB .
Justify your conjecture.
Exploring a Segment Bisector
Work with a partner. Fold and crease a piece
of paper, as shown. Label the ends of the crease
as A and B.
a. Fold the paper again so that point A coincides
with point B. Crease the paper on that fold.
b. Unfold the paper and examine the four angles
formed by the two creases. What can you
conclude about the four angles?
Writing a Conjecture
Work with a partner.
a. Draw
—
AB , as shown.
b. Draw an arc with center A on each
side of
—
AB . Using the same compass
setting, draw an arc with center B
on each side of
—
AB . Label the
intersections of the arcs C and D.
c. Draw
—
CD . Label its intersection
with
—
AB as O. Write a conjecture
about the resulting diagram. Justify
your conjecture.
Communicate Your AnswerCommunicate Your Answer
4. What conjectures can you make about perpendicular lines?
5. In Exploration 3, nd AO and OB when AB = 4 units.
CONSTRUCTING
VIABLE
ARGUMENTS
To be pro cient in math,
you need to make
conjectures and build a
logical progression of
statements to explore the
truth of your conjectures.
Essential QuestionEssential Question What conjectures can you make about
perpendicular lines?
BAO
C
D
A
B
O
A
B
DC
A
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148 Chapter 3 Parallel and Perpendicular Lines
3.4
Lesson
What You Will LearnWhat You Will Learn
Find the distance from a point to a line.
Construct perpendicular lines.
Prove theorems about perpendicular lines.
Solve real-life problems involving perpendicular lines.
Finding the Distance from a Point to a Line
The distance from a point to a line is the length of the perpendicular segment from
the point to the line. This perpendicular segment is the shortest distance between the
point and the line. For example, the distance between point A and line k is AB.
k
A
B
distance from a point to a line
Finding the Distance from a Point to a Line
Find the distance from point A to
⃖
""
⃗
BD .
x
y
4
−4
4
−4
A(−3, 3)
B(−1, −3)
C(1, −1)
D(2, 0)
SOLUTION
Because
—
AC ⊥
⃖
""
⃗
BD , the distance from point A to
⃖
""
⃗
BD is AC. Use the Distance Formula.
AC =
√
———
(−3 − 1)
2
+ [3 − (−1)]
2
=
√
—
(−4)
2
+ 4
2
=
√
—
32 ≈ 5.7
So, the distance from point A to
⃖
""
⃗
BD is about 5.7 units.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
1. Find the distance from point E to
⃖
""
⃗
FH .
x
y
4
−2
2−4
E(−4, −3)
F(0, 3)
G(1, 2)
H(2, 1)
REMEMBER
Recall that if A(x
1
, y
1
)
and C(x
2
, y
2
) are points
in a coordinate plane,
then the distance
between A and C is
AC =
√
———
(x
2
− x
1
)
2
+ (y
2
− y
1
)
2
.
distance from a point to a line,
p. 148
perpendicular bisector, p. 149
Core Vocabulary
Core Vocabu
l
l
a
r
r
y
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Section 3.4 Proofs with Perpendicular Lines 149
Constructing Perpendicular Lines
Constructing a Perpendicular Line
Use a compass and straightedge to construct a line
perpendicular to line m through point P, which is
not on line m.
SOLUTION
Step 1 Step 2 Step 3
A
B
m
P
P
P
A
B
Q
m
P
Q
AB
Q
m
P
Draw arc with center P Place the
compass at point P and draw an arc
that intersects the line twice. Label
the intersections A and B.
Draw intersecting arcs Draw an
arc with center A. Using the same
radius, draw an arc with center B.
Label the intersection of the arcs Q.
Draw perpendicular line Draw
⃖
""
⃗
PQ . This line is perpendicular to
line m.
Constructing a Perpendicular Bisector
Use a compass and straightedge to construct
the perpendicular bisector of
—
AB .
SOLUTION
Step 1 Step 2 Step 3
BA
BA
BMA
in. 1 2 3 4 5 6
cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Draw an arc Place the compass
at A. Use a compass setting that is
greater than half the length of
—
AB .
Draw an arc.
Draw a second arc Keep the same
compass setting. Place the compass
at B. Draw an arc. It should intersect
the other arc at two points.
Bisect segment Draw a line
through the two points of
intersection. This line is the
perpendicular bisector of
—
AB . It
passes through M, the midpoint
of
—
AB . So, AM = MB.
P
m
The perpendicular bisector of a line segment
—
PQ is the line n with the following
twoproperties.
• n ⊥
—
PQ
• n passes through the midpoint M of
—
PQ .
M
n
QP
BA
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150 Chapter 3 Parallel and Perpendicular Lines
Proving the Perpendicular Transversal Theorem
Use the diagram to prove the
Perpendicular Transversal Theorem.
SOLUTION
Given h ( k, j ⊥ h
Prove j ⊥ k
STATEMENTS REASONS
1. h ( k, j ⊥ h 1. Given
2. m∠2 = 90° 2. De nition of perpendicular lines
3. ∠2 ≅ ∠6 3. Corresponding Angles Theorem (Theorem 3.1)
4. m∠2 = m∠6 4. De nition of congruent angles
5. m∠6 = 90° 5. Transitive Property of Equality
6. j ⊥ k 6. De nition of perpendicular lines
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
2. Prove the Perpendicular Transversal Theorem using the diagram in Example 2
and the Alternate Exterior Angles Theorem (Theorem 3.3).
Proving Theorems about Perpendicular Lines
Theorem 3.10 Linear Pair Perpendicular Theorem
If two lines intersect to form a linear pair of
congruent angles, then the lines are perpendicular.
If ∠l ≅ ∠2, then g ⊥ h.
Proof Ex. 13, p. 153
Theorem 3.11 Perpendicular Transversal Theorem
In a plane, if a transversal is perpendicular to one
of two parallel lines, then it is perpendicular to the
other line.
If h ( k and j ⊥ h, then j ⊥ k.
Proof Example 2, p. 150; Question 2, p. 150
Theorem 3.12 Lines Perpendicular to a Transversal Theorem
In a plane, if two lines are perpendicular to the
same line, then they are parallel to each other.
If m ⊥ p and n ⊥ p, then m ( n.
Proof Ex. 14, p. 153; Ex. 47, p. 162
TheoremsTheorems
h
g
12
h
k
j
p
nm
h
j
k
1
2
43
5
6
87
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Section 3.4 Proofs with Perpendicular Lines 151
Solving Real-Life Problems
Proving Lines Are Parallel
The photo shows the layout of a neighborhood. Determine which lines, if any, must
be parallel in the diagram. Explain your reasoning.
s t u
p
q
stu
p
q
SOLUTION
Lines p and q are both perpendicular to s, so by the Lines Perpendicular to a
Transversal Theorem, p ( q. Also, lines s and t are both perpendicular to q, so by the
Lines Perpendicular to a Transversal Theorem, s ( t.
So, from the diagram you can conclude p ( q and s ( t.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
Use the lines marked in the photo.
ab
c
d
3. Is b ( a? Explain your reasoning.
4. Is b ⊥ c? Explain your reasoning.
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152 Chapter 3 Parallel and Perpendicular Lines
Exercises
3.4
Dynamic Solutions available at BigIdeasMath.com
In Exercises 3 and 4, nd the distance from point A
to
⃖
""
⃗
XZ . (See Example 1.)
3.
x
y
4
6
4
−2
X(−1, −2)
Y(0, 1)
Z(2, 7)
A(3, 0)
4.
x
y
Z(4, −1)
Y(2, −1.5)
X(−4, −3)
A(3, 3)
3
1
−3
−1
1−3
CONSTRUCTION In Exercises 5–8, trace line m and point
P. Then use a compass and straightedge to construct a
line perpendicular to line m through point P.
5.
m
P
6.
m
P
7.
m
P
8.
m
P
CONSTRUCTION In Exercises 9 and 10, trace
—
AB . Then
use a compass and straightedge to construct the
perpendicular bisector of
—
AB .
9.
A
B
10.
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. COMPLETE THE SENTENCE The perpendicular bisector of a segment is the line that passes through
the________ of the segment at a ________ angle.
2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Find XZ.
Find the distance from point X to line
⃖
""
⃗
WZ .
Find the length of
—
XY .
Find the distance from line to point X.
x
y
4
−4
−2
4
2−2−4
X(−3, 3)
W(2, −2)
Z(4, 4)
Y(3, 1)
Vocabulary and Core Concept Check
A
B
p
p
p
p
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Section 3.4 Proofs with Perpendicular Lines 153
ERROR ANALYSIS In Exercises 11 and 12, describe and
correct the error in the statement about the diagram.
11.
x
zy
Lines y and z are parallel.
✗
12.
B
8 cm
12 cm
C
A
The distance from point C to
⃖
""
⃗
AB
is 12 centimeters.
✗
PROVING A THEOREM In Exercises 13 and 14, prove the
theorem. (See Example 2.)
13. Linear Pair Perpendicular Theorem (Thm. 3.10)
14. Lines Perpendicular to a Transversal Theorem
(Thm. 3.12)
PROOF In Exercises 15 and 16, use the diagram to write
a proof of the statement.
15. If two intersecting lines are perpendicular, then they
intersect to form four right angles.
Given a ⊥ b
Prove ∠1, ∠2, ∠3, and ∠4 are right angles.
a
b
12
43
16. If two sides of two adjacent acute angles are
perpendicular, then the angles are complementary.
Given
"
""
⃗
BA ⊥
"
""
⃗
BC
Prove ∠1 and ∠2 are complementary.
A
B
1
2
C
In Exercises 17–22, determine which lines, if any, must
be parallel. Explain your reasoning. (See Example 3.)
17.
x
y
wv
18.
19.
p
q
nm
20.
21. 22.
23.
USING STRUCTURE Find all the unknown angle
measures in the diagram. Justify your answer for each
angle measure.
1
30°
40°
2
3
4
5
24. MAKING AN ARGUMENT Your friend claims that
because you can nd the distance from a point to a
line, you should be able to nd the distance between
any two lines. Is your friend correct? Explain
your reasoning.
25. MATHEMATICAL CONNECTIONS Find the value of x
when a⊥ b and b ( c.
c
a
b
(9x + 18)°
[5(x + 7) + 15]°
b
c
a
c
d
ba
k
m
np
x
w
v
zy
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154 Chapter 3 Parallel and Perpendicular Lines
26. HOW DO YOU SEE IT? You are trying to cross a
stream from point A. Which point should you jump
to in order to jump the shortest distance? Explain
your reasoning.
27. ATTENDING TO PRECISION In which of the following
diagrams is
—
AC (
—
BD and
—
AC ⊥
—
CD ? Select all
that apply.
A
C
A
B
D
B
C
A
D
C
B
A
C
D
D
C
AB
D
E
C
A
B
D
28. THOUGHT PROVOKING The postulates and theorems
in this book represent Euclidean geometry. In
spherical geometry, all points are points on the surface
of a sphere. A line is a circle on the sphere whose
diameter is equal to the diameter of the sphere. In
spherical geometry, how many right angles are formed
by two perpendicular lines? Justify your answer.
29. CONSTRUCTION Construct a square of side
length AB.
BA
30. ANALYZING RELATIONSHIPS The painted line
segments that form the path of a crosswalk are usually
perpendicular to the crosswalk. Sketch what the
segments in the photo would look like if they were
perpendicular to the crosswalk. Which type of line
segment requires less paint? Explain your reasoning.
31. ABSTRACT REASONING Two lines, a and b, are
perpendicular to line c. Line d is parallel to line c.
The distance between lines a and b is x meters. The
distance between lines c and d is y meters. What
shape is formed by the intersections of the four lines?
32. MATHEMATICAL CONNECTIONS Find the distance
between the lines with the equations y =
3
—
2
x + 4 and
−3x + 2y = −1.
33. WRITING Describe how you would nd the distance
from a point to a plane. Can you nd the distance
from a line to a plane? Explain your reasoning.
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Simplify the ratio. (Skills Review Handbook)
34.
6 − (−4)
—
8 − 3
35.
3 − 5
—
4 − 1
36.
8 − (−3)
—
7 − (−2)
37.
13 − 4
—
2 − (−1)
Identify the slope and the y-intercept of the line.
(Skills Review Handbook)
38. y = 3x + 9 39. y = −
1
—
2
x + 7
40. y =
1
—
6
x − 8
41. y = −8x − 6
Reviewing what you learned in previous grades and lessons
y
our reason
i
ng
g
.
BCD
A
E
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