2.2 Angle Bisectors - Murrieta Valley Unified School District

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 5 1108.

BD

A

Find m¡ÏABD and m¡ÏDBC.

D

1108

B

Solution

1

2

1

5 }}(1108)

2

&( bisects aABC.

BD

5 558

Simplify.

maABD 5 }}(maABC )

C

Substitute 1108 for maABC.

aABD and aDBC are congruent, so maDBC 5 maABD.

ANSWER

? So, maABD 5 558, and maDBC 5 558.

Find Angle Measures

&*( bisects aGHJ. Find maGHK and maKHJ.

HK

1.

2.

3.

G

K

K

G

K

J

1618

528

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 5 468.

MP

2

Find Angle Measures and Classify an Angle

L

a. Find maPMN and maLMN.

b. Determine whether aLMN is acute,

P

468

right, obtuse, or straight. Explain.

H

G

M

Solution

J

E

F

N

&*( bisects aLMN, so maLMP 5 maPMN.

a. MP

K

You know that maLMP 5 468. Therefore, maPMN 5 468.

aEFH c aHFG

aGFJ c aJFK

The measure of aLMN is twice the measure of aLMP.

maLMN 5 2(maLMP) 5 2(468) 5 928

So, maPMN 5 468, and maLMN 5 928.

b. aLMN is obtuse because its measure is between 908 and 1808.

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

298

P

P

S

S

608

P

P

458

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.

458

B

D

Solution

maDAB 5 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

5 2(458)

Substitute 458 for maBAC.

5 908

Simplify.

maBCD 5 2(maACB)

ANSWER

Segments and Angles

&( bisects aDAB.

AC

&( bisects aBCD.

CA

5 2(278)

Substitute 278 for maACB.

5 548

Simplify.

? The measure of aDAB is 908, and the measure of

aBCD is 548.

278

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

968

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 1)8

P

858

R

IStudent Help

S

Solution



MORE EXAMPLES

More examples at



maPRQ 5 maQRS

&*( bisects aPRS.

RQ

(6x 1 1)8 5 858

Substitute given measures.

6x 1 1 2 1 5 85 2 1

6x 5 84

Simplify.

6x

84

}} 5 }}

6

6

Divide each side by 6.

x 5 14

CHECK

Subtract 1 from each side.

Simplify.

? You can check your answer by substituting 14 for x.

maPRQ 5 (6x 1 1)8 5 (6 p 14 1 1)8 5 (84 1 1)8 5 858

Use Algebra with Angle Measures

&( bisects aABC. Find the value of x.

BD

9.

B

558

C

A

10.

A

D

(x 1 12)8

9x 8

(8x 1 3)8

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

1688

828

408

C

7. Find maABC.

B

D

A

B

C

308

A

398

508

B

B

D

A

C

Practice and Applications

&( bisects aPQR. Find maPQS and

Finding Angle Measures QS

maSQR.

Extra Practice

See p. 677.

8.

9.

P

10.

P

S

P

S

P

1248

508

P

1088

P

R

11.

12.

R

13.

S

S

R

S

R

S

P

P

678

918

P

P

P

758

R

R

P

&( bisects aPQR. Find maPQS and maSQR.

Fans QS

Homework Help

Example 1:

Example 2:

Example 3:

Example 4:

Exs. 8¨C16

Exs. 17¨C22

Exs. 24¨C27

Exs. 28¨C30

S

14.

P

Chapter 2

S

16.

S

R

878

Segments and Angles

?

R

P

1808

P

?

64

15.

R

1208

?

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

C

D

458

C

228

C

B

20.

A

B

D

21.

B

808

388

A

22.

C

D

C

178

D

738

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¨C27, use the diagram below. When light is

reflected by a smooth surface, the angle of incidence is equal to the

angle of reflection.

J

R Angle of

reflection

Angle of

incidence

lig

ht

wa

ve

s

P

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.

25. If the angle of reflection is 678, what is the angle of incidence?

26. If maJKL 5 1098, what is the angle of reflection?

27.

Application Links



Can you determine whether aJKP is

congruent to aLKQ in the diagram above? Explain your reasoning.

You be the Judge

2.2

Angle Bisectors

65

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