2.2 Angle Bisectors

Page 1 of 6

2.2 Angle Bisectors

Goal

Bisect an angle.

Key Words

? angle bisector

An angle bisector is a ray that divides an angle into two angles that are congruent. In the photograph of the hang glider, B&D*( bisects aABC because it divides the angle into two congruent angles.

Visualize It!

In this book, an arc that crosses two or more angles identifies the measure of the entire angle it crosses.

aABD c aDBC

If B&D*( bisects aABC, then the measures of aABD and aDBC are half the measure of aABC. Also, the measure of aABC is twice the measure of aABD or aDBC .

EXAMPLE 1 Find Angle Measures

B&D*( bisects a ABC, and ma ABC 110.

Find mABD and mDBC.

A D

110

Solution

B

C

ma ABD 12(ma ABC )

B&D( bisects aABC.

12(110)

Substitute 110 for maABC.

55

Simplify.

aABD and aDBC are congruent, so maDBC maABD.

ANSWER So, ma ABD 55, and maDBC 55.

Find Angle Measures

H&K*( bisects aGHJ. Find maGHK and maKHJ.

1.

K

G

52

J

H

2. G

K J

H

3. J

K 161 HG

2.2 Angle Bisectors

61

Page 2 of 6

Visualize It!

In this book, matching red arcs identify congruent angles in diagrams.

H

G

E

J

F

K

aEFH c aHFG aGFJ c aJFK

EXAMPLE 2 Find Angle Measures and Classify an Angle

M&P*( bisects aLMN, and maLMP 46. a. Find maPMN and maLMN. b. Determine whether aLMN is acute, right, obtuse, or straight. Explain.

L 46 P

M

N

Solution a. M&P*( bisects aLMN, so maLMP maPMN. You know that maLMP 46. Therefore, maPMN 46. 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

Q&S( bisects aPQR. Find maSQP and maPQR. Then determine whether aPQR is acute, right, obtuse, or straight.

4. R 29

S P

P

5.

S

P

45

R

P

6.

P

S

60

P

R

Careers

KITE DESIGNERS use geometric principles in designing and making kites. A kite's struts often bisect the angles they support.

EXAMPLE 3 Use Angle Bisectors

A

In the kite, aDAB is bisected by A&C(,

45

and aBCD is bisected by C&A(.

Find maDAB and maBCD.

D

Solution

maDAB 2(maBAC ) A&C( bisects aDAB.

2(45)

Substitute 45 for maBAC.

90

Simplify.

maBCD 2(ma ACB) C&A( bisects aBCD.

2(27)

Substitute 27 for maACB.

54

Simplify.

ANSWER The measure of aDAB is 90, and the measure of aBCD is 54.

B

27 C

62

Chapter 2 Segments and Angles

Page 3 of 6

Use Angle Bisectors

7. K&M*( bisects aJKL. Find maJKM and maMKL.

8. U&V*( bisects aWUT. Find maWUV and maWUT.

K 96

J

W L

U T

V M

IStudent Help



MORE EXAMPLES More examples at

EXAMPLE 4 Use Algebra with Angle Measures

R&Q( bisects aPRS. Find the value of x. P

Solution

(6x 1)

P

85

RS

maPRQ maQRS R&Q*( bisects aPRS.

(6x 1) 85

Substitute given measures.

6x 1 1 85 1

Subtract 1 from each side.

6x 84

Simplify.

66x 864

Divide each side by 6.

x 14

Simplify.

CHECK 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

B&D( bisects aABC. Find the value of x.

9. C

B

A

(x 12) 55

D

10. B

A

D

9x (8x 3)

C

2.2 Angle Bisectors

63

Page 4 of 6

2.2 Exercises

Guided Practice

Vocabulary Check 1. What kind of geometric figure is an angle bisector?

Skill Check

B&D( bisects aABC. Find the angle measure.

2. Find ma ABD.

3. Find maDBC.

A

D

40

B

C

D

C

82

A

B

5. Find ma ABC.

6. Find maCBA.

D A

C 30

B

A

D

50

B

C

4. Find maDBC.

D

168 A BC

7. Find ma ABC.

B

C

39 D A

Practice and Applications

Extra Practice

See p. 677.

Finding Angle Measures Q&S( bisects aPQR. Find maPQS and maSQR.

8.

P

S P

50

R

9.

P

S

108

P

R

10. S

P

124 P R

11. S

P

R 67

P

12. S

R

13.

P

P

S

75 P

91

P

R

Homework Help

Fans Q&S( bisects aPQR. Find maPQS and maSQR.

14.

S

15.

S

16.

Example 1: Exs. 8?16 Example 2: Exs. 17?22 Example 3: Exs. 24?27 Example 4: Exs. 28?30

P

87 OE

R P

180

P

OE

R

64

Chapter 2 Segments and Angles

S

R 120 OE

Page 5 of 6

Finding Angle Measures B&D*( bisects aABC. Find maABD and maABC. Then determine whether aABC is acute, right, obtuse, or straight.

17.

A

D

22

B

C

18. C

B 45

D

19. A

A

D

C

38

B

20. A

B C

80

D

21.

C

17 D

A

B

22. A

D

73 C B

Science

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 A&K(.

A

P

B

N

C

M

D

R

Q

L

E

K

F

JHG

LASERS The photograph above shows a student using lasers as part of his research in optical computing.

Application Links



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.

Angle of R Angle of

incidence J

light waves

reflection L

P

K

P

24. Name an angle bisector in the diagram.

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

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download