Page 1 of 6
2.2
Goal
Bisect an angle.
Angle Bisectors
An angle bisector is a ray that divides an angle into two angles
&*( bisects
that are congruent. In the photograph of the hang glider, BD
aABC because it divides the angle into two congruent angles.
Key Words
• angle bisector
aABD c aDBC
&*( bisects aABC, then the measures of aABD and aDBC are half the
If BD
measure of aABC. Also, the measure of aABC is twice the measure of
aABD or aDBC .
Visualize It!
In this book, an arc that
crosses two or more
angles identifies the
measure of the entire
angle it crosses.
EXAMPLE
1
Find Angle Measures
&*( bisects aABC, and maABC 110.
BD
A
Find m∠ABD and m∠DBC.
D
110
B
Solution
1
2
1
(110)
2
&( bisects aABC.
BD
55
Simplify.
maABD (maABC )
C
Substitute 110 for maABC.
aABD and aDBC are congruent, so maDBC maABD.
ANSWER
So, maABD 55, and maDBC 55.
Find Angle Measures
&*( bisects aGHJ. Find maGHK and maKHJ.
HK
1.
2.
3.
G
K
K
G
K
J
161
52
J
J
H
H
H
2.2
Angle Bisectors
G
61
Page 2 of 6
Visualize It!
EXAMPLE
In this book, matching
red arcs identify
congruent angles in
diagrams.
&*( bisects aLMN, and maLMP 46.
MP
2
Find Angle Measures and Classify an Angle
L
a. Find maPMN and maLMN.
b. Determine whether aLMN is acute,
P
46
right, obtuse, or straight. Explain.
H
G
M
Solution
J
E
F
N
&*( bisects aLMN, so maLMP maPMN.
a. MP
K
You know that maLMP 46. Therefore, maPMN 46.
aEFH c aHFG
aGFJ c aJFK
The measure of aLMN is twice the measure of aLMP.
maLMN 2(maLMP) 2(46) 92
So, maPMN 46, and maLMN 92.
b. aLMN is obtuse because its measure is between 90 and 180.
Find Angle Measures and Classify an Angle
&( bisects aPQR. Find maSQP and maPQR. Then determine
QS
whether aPQR is acute, right, obtuse, or straight.
4.
5.
R
S
29
P
P
S
S
60
P
P
45
P
Careers
6.
R
EXAMPLE
3
R
P
Use Angle Bisectors
A
&(,
In the kite, aDAB is bisected by AC
&(
and aBCD is bisected by CA .
Find maDAB and maBCD.
45
B
D
Solution
maDAB 2(maBAC)
KITE DESIGNERS use
geometric principles in
designing and making kites.
A kite’s struts often bisect
the angles they support.
62
Chapter 2
2(45)
Substitute 45 for maBAC.
90
Simplify.
maBCD 2(maACB)
ANSWER
Segments and Angles
&( bisects aDAB.
AC
&( bisects aBCD.
CA
2(27)
Substitute 27 for maACB.
54
Simplify.
The measure of aDAB is 90, and the measure of
aBCD is 54.
27
C
Page 3 of 6
Use Angle Bisectors
&*( bisects aJKL.
7. KM
&*( bisects aWUT.
8. UV
Find maJKM and maMKL.
Find maWUV and maWUT.
K
96
U
W
J
T
L
V
M
EXAMPLE
4
Use Algebra with Angle Measures
&( bisects aPRS. Find the value of x.
RQ
P
(6x 1)
P
85
R
IStudent Help
S
Solution
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
maPRQ maQRS
&*( bisects aPRS.
RQ
(6x 1) 85
Substitute given measures.
6x 1 1 85 1
6x 84
Simplify.
6x
84
6
6
Divide each side by 6.
x 14
CHECK
Subtract 1 from each side.
Simplify.
✓ You can check your answer by substituting 14 for x.
maPRQ (6x 1) (6 p 14 1) (84 1) 85
Use Algebra with Angle Measures
&( bisects aABC. Find the value of x.
BD
9.
B
A
10.
A
(x 12)
55
C
D
9x (8x 3)
D
B
C
2.2
Angle Bisectors
63
Page 4 of 6
2.2 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. What kind of geometric figure is an angle bisector?
&( bisects aABC. Find the angle measure.
BD
2. Find maABD.
3. Find maDBC.
A
D
4. Find maDBC.
C
D
D
B
A
C
5. Find maABC.
A
B
6. Find maCBA.
C
D
168
82
40
30
A
B
C
39
50
D
A
B
B
C
7. Find maABC.
D
A
B
C
Practice and Applications
&( bisects aPQR. Find maPQS and
Finding Angle Measures QS
maSQR.
Extra Practice
See p. 677.
8.
9.
P
P
S
P
10.
124
50
P
108
P
R
11.
13.
S
S
P
P
67
R
S
R
12.
R
S
P
S
91
P
P
P
75
R
R
P
&( bisects aPQR. Find maPQS and maSQR.
Fans QS
Homework Help
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 8–16
Exs. 17–22
Exs. 24–27
Exs. 28–30
S
14.
P
Chapter 2
S
16.
S
R
P
87
Segments and Angles
Œ
R
P
180
Œ
64
15.
R
120
Œ
Page 5 of 6
&*( bisects aABC. Find maABD and
Finding Angle Measures BD
maABC. Then determine whether aABC is acute, right, obtuse, or
straight.
17.
18.
A
19.
B
A
D
45
C
22
20.
A
B
D
21.
B
80
38
A
C
B
C
D
22.
C
D
C
17
D
73
D
A
23. Paper Airplanes The diagram at
the right represents an unfolded
piece of paper used to make a
paper airplane like the one shown
below. Using the diagram, name the
&(.
angles that are bisected by AK
A
B
B
A
P
C
B
C
N
M
D
R
Q
L
E
K
F
Science
J
H
G
Lasers In Exercises 24–27, use the diagram below. When light is
reflected by a smooth surface, the angle of incidence is equal to the
angle of reflection.
J
P
R Angle of
reflection
Angle of
incidence
lig
ht
wa
ve
s
K
L
P
24. Name an angle bisector in the diagram.
LASERS The photograph
above shows a student using
lasers as part of his research
in optical computing.
Application Links
CLASSZONE.COM
25. If the angle of reflection is 67, what is the angle of incidence?
26. If maJKL 109, what is the angle of reflection?
27.
You be the Judge Can you determine whether aJKP is
congruent to aLKQ in the diagram above? Explain your reasoning.
2.2
Angle Bisectors
65
Page 6 of 6
&*( bisects aJKL. Find the value of the variable.
Using Algebra KM
28.
M
(y 29)
29.
30.
J
J
25
(3z 4)
54
J
K
(8x 13)
M
L
K
L
75
L
M
K
31. Technology Use geometry software
to draw T ABC. Construct the angle
bisector of aBAC. Then find the midpoint
&*. Drag any of the points. Does the
of BC
angle bisector always pass through the
midpoint of the opposite side? Does it
ever pass through the midpoint?
B
D
&*( bisects aABE and
32. Challenge BD
F
100
A
Mixed Review
E
D
&( bisects aEBC. Use the diagram
BF
shown to find maDBF.
Standardized Test
Practice
C
A
40
B
C
&*( bisects aABC
33. Multiple Choice In the diagram below, BD
and maABC 23. What is maABD?
A
C
11.5
23
B
D
A
12.5
D
46
B
C
Describing Number Patterns Describe a pattern in the numbers.
Write the next number you expect in the pattern. (Lesson 1.1)
34. 1, 9, 17, 25, . . .
35. 13, 8, 3, 2, . . .
36. 5.6, 5.16, 5.116, 5.1116, . . .
37. 60, 30, 15, 7.5, . . .
Classifying Angles Use the diagram below to classify the angle
as acute, right, obtuse, or straight. (Lesson 1.6)
38. aEBC
39. aABE
40. aDBC
41. aABC
D
30
A
Algebra Skills
66
Chapter 2
E
B
C
Solving Equations Solve the equation. (Skills Review, p. 673)
42. 2x 15 9
43. 3a 12 48
44. 10 3y 52
45. 5m 11 46
46. 2z 4 8
47. 3 n 23
Segments and Angles