Page 1 of 7
3.2
Goal
Theorems About
Perpendicular Lines
Geo-Activity
Use theorems about
perpendicular lines.
1 Fold a piece of
●
Key Words
• complementary angles
p. 67
paper to form
a line.
Intersecting Lines
2 Fold the paper
●
3 Unfold the paper.
●
again by lining
up the first fold.
Label the angles
as shown.
• perpendicular lines
p. 108
4 1
3 2
4 Do the fold lines appear perpendicular? Use a protractor to measure
●
each angle. How many right angles are formed?
5 Write one or more conjectures about perpendicular lines.
●
The Geo-Activity above suggests the following theorems about
perpendicular lines.
THEOREMS 3.1 and 3.2
Student Help
Theorem 3.1
STUDY TIP
Words
All right angles are congruent.
Symbols
If maA 90 and maB 90,
then aA c aB.
B
A
Theorem 3.2
Theorem 3.2 tells you
that if one right angle
is marked on a pair of
intersecting lines, then
the other three angles
are also right angles.
114
Chapter 3
Words
n
If two lines are perpendicular, then
they intersect to form four right angles.
Symbols
If n ∏ m, then ma1 90, ma2 90,
ma3 90, and ma4 90.
Parallel and Perpendicular Lines
1
4
3
2
m
Page 2 of 7
1
EXAMPLE
Perpendicular Lines and Reasoning
In the diagram, r ∏ s and r ∏ t. Determine whether enough
information is given to conclude that the statement is true.
Explain your reasoning.
r
1 2
a. a3 c a5
u
3
b. a4 c a5
5
c. a2 c a3
4
t
s
Solution
a. Yes, enough information is given. Both angles are right
angles. By Theorem 3.1, they are congruent.
b. Yes, enough information is given. Lines r and t are
perpendicular. So, by Theorem 3.2, a4 is a right angle.
By Theorem 3.1, all right angles are congruent.
c. Not enough information is given to conclude that a2 c a3.
Perpendicular Lines and Reasoning
In the diagram, g ∏ e and g ∏ f. Determine whether enough
information is given to conclude that the statement is true. Explain.
1. a6 c a10
3. a6 c a8
2. a7 c a10
4. a7 c a11
g
h
6
8
9
7
5. a7 c a9
6. a6 c a11
10
e
f
11
Student Help
LOOK BACK
Theorems 3.3 and 3.4
refer to adjacent angles.
For the definition of
adjacent angles, see
p. 68.
THEOREMS 3.3 and 3.4
Theorem 3.3
Words
If two lines intersect to form
adjacent congruent angles, then
the lines are perpendicular.
Symbols
B
A
1
D
2
^&( ∏ BD
^&(.
If a1 c a2, then AC
C
Theorem 3.4
Words
If two sides of adjacent acute angles
are perpendicular, then the angles
are complementary.
Symbols
&( ∏ EH
&(, then ma3 ma4 90.
If EF
3.2
G
F
3
4
E
H
Theorems About Perpendicular Lines
115
Page 3 of 7
2
EXAMPLE
Aviation
Use Theorems About Perpendicular Lines
In the helicopter at the right, are
aAXB and aCXB right angles? Explain.
B
C
Solution
If two lines intersect to form adjacent
&* and BD
&* do,
congruent angles, as AC
then the lines are perpendicular
&* ∏ BD
&*.
(Theorem 3.3). So, AC
A
X
D
&* and BD
&* are perpendicular,
Because AC
they form four right angles (Theorem 3.2).
So, aAXB and aCXB are right angles.
HELICOPTERS Main rotors
of a helicopter may have two
to eight blades. The blades
create the helicopter’s lift
power.
3
EXAMPLE
Use Algebra with Perpendicular Lines
In the diagram at the right,
F
&*( ∏ EH
&*( and maGEH 30.
EF
Find the value of y.
E
6y G
30
H
Solution
&*( ∏ EH
&*(.
aFEG and aGEH are adjacent acute angles and EF
So, aFEG and aGEH are complementary (Theorem 3.4).
6y 30 90
6y 60
y 10
ANSWER
maFEG maGEH 90
Subtract 30 from each side.
Divide each side by 6.
The value of y is 10.
Use Algebra with Perpendicular Lines
Find the value of the variable. Explain your reasoning.
7. aEFG c aHFG
&( ∏ AD
&*(
8. AB
&( ∏ KL
&(,
9. KJ
aJKM c aMKL
G
B
5x E
F
H
Chapter 3
Parallel and Perpendicular Lines
J
M
36
9y A
116
C
z
D
L
z
K
Page 4 of 7
3.2 Exercises
Guided Practice
Vocabulary Check
1. Complete the statement: If two lines intersect to form adjacent
congruent angles, then the lines are __?__.
Skill Check
Write the theorem that justifies the statement about the diagram.
2. a5 and ∠6 are
3. j ∏ k
4. ma9 ma10 90
right angles.
l
j
g
f
5
7
m
8
k
6
9
h
10
In Exercises 5–7, p ∏ q. Write an equation to find the value of x.
(Do not solve the equation.)
p
5.
p
6.
x
45
q
x
p
7.
70
q
x
q
Practice and Applications
Extra Practice
See p. 679.
Perpendicular Lines and Reasoning Determine whether enough
information is given to conclude that the statement is true. Explain.
8. a2 c a5
c
9. a6 c a7
1
4 3
10. a1 c a3
11. a1 c a5
5
2
6
7
d
c∏d
Logical Reasoning What can you conclude about a1 and a2 using
the given information?
&( ∏ BC
^&(
12. BA
Homework Help
13. n ∏ m
14. h ∏ k
h
n
A
Example 1: Exs. 8–11
Example 2: Exs. 12–14
Example 3: Exs. 17–22
D
2
B
1 2
1
m
1
2
k
C
3.2
Theorems About Perpendicular Lines
117
Page 5 of 7
Error Analysis Students were asked to set up an equation to find
the value of x, given that v ∏ w. Describe and correct any errors.
15.
16.
v
56
v
9x
(x + 4)
w
w
(x + 4)° + 90° = 56°
9x° = 180°
Using Algebra Find the value of x, given that s ∏ t.
IStudent Help
ICLASSZONE.COM
s
17.
HOMEWORK HELP
Extra help with problem
solving in Exs. 17–22 is
at classzone.com
s
18.
t
x
s
20.
(2x 20)
60
x
s
21.
s
19.
x
55
t
s
22.
5x (3x 5)
40
t
t
10x t
t
^&( ∏ BD
^&(. Then use
Angle Measures Find the value of x, given that AB
the value of x to find maCBD.
23.
2x 24.
25.
C
C
A
7x B
8x D
(x 20)
A
C
x
10x D
A
D
B
B
26. Window Repair You are fixing a window
frame. You fit two strips of wood together
to make the crosspieces. For the glass
panes to fit, each angle formed by the
crosspieces must be a right angle. Do you
need to measure all four angles to be sure
they are all right angles? Explain.
27.
You be the Judge
In the diagram
shown, a1 and a3 are congruent and
complementary. Can you conclude that
&*( ∏ BC
&*(? Explain your reasoning.
BA
A
C
1
2 3
B
118
Chapter 3
Parallel and Perpendicular Lines
4
Page 6 of 7
Sports
Sports In orienteering, a compass and a map are used to navigate
through a wilderness area. Suppose you are in an orienteering event
and you are traveling at 40 east of magnetic north, as shown below.
N
40
y
W
E
ORIENTEERING
The students above are part
of an in-school orienteering
training program at Clark
Montessori High School in
Cincinnati, Ohio.
S
28. What can you conclude, given that ∠NYW and ∠SYW are
congruent? Explain.
29. How many degrees do you need to turn to travel due east?
30. How many degrees do you need to turn to travel due south
from the position shown on the compass?
Origami Origami is the Japanese art of folding pieces of paper into
objects. The folds on the paper shown below are the basis for many
&* ∏ HD
&*.
objects. On the paper, BF
31. Are aDJE and aEJF complementary?
A
B
C
Explain your reasoning.
32. If maBJC maCJD, what are
H
their measures?
D
J
33. Is there enough information to
conclude that aAJG is a right angle?
Explain your reasoning.
Standardized Test
Practice
G
F
E
Multiple Choice In Exercises 34 and 35, use the diagram below.
34. Which of the following is true if g ∏ h?
A
B
C
D
ma1 ma2 > 180
g
ma1 ma2 < 180
1
2
ma1 ma2 180
h
None of these
35. If g ∏ h and ma1 40, what is ma2?
F
40
G
H
50
3.2
60
J
140
Theorems About Perpendicular Lines
119
Page 7 of 7
Mixed Review
Classifying Angles State whether the angle appears to be acute,
right, obtuse, or straight. Then estimate its measure. (Lesson 1.6)
36.
37.
38.
Finding Complements and Supplements Find the measure of the
angle. (Lesson 2.3)
39. aA is a complement of aB, and maA 37. Find maB.
40. aC is a supplement of aD, and maC 56. Find maD.
Vertical Angles Find the value of x. (Lesson 2.4)
41.
42.
80
150 (9x 30)
(4x 20)
Algebra Skills
43.
12x (10x 6)
Decimals Evaluate. (Skills Review, p. 655)
44. 13.6 9.8
45. 14 2.21
46. 7.4 5.9
47. 79.2 9
48. 100 4.5 26.1
49. 30 11.1
Quiz 1
Think of each segment on the shopping bag as part of a line. There
may be more than one correct answer. (Lesson 3.1)
1. Name two lines perpendicular
^&*(.
to FG
E
A
F
^&(.
2. Name a line skew to BF
B
3. Name a line that appears parallel
^&(.
to AD
H
4. Name a line perpendicular to
plane HGC.
G
D
C
Find the value of the variable, given that p ∏ q. (Lesson 3.2)
5.
6.
p
z
57
23
q
120
Chapter 3
Parallel and Perpendicular Lines
7.
p
p
(3y 12)
3x q
q