Deductive Reasoning

“The proof is in the pudding.”

“Indubitably.”Je solve le crime. Pompt de pompt pompt."

Le pompt de pompt le solve de crime!"

2-4 Special Pairs of Angles WE

2-4 Written ExercisesDetermine the measures of the complements and supplement of each angle. measure sums up to 900.

1 20m K

Complementary Supplementary

90 – 20 = 70 180 – 10 = 170

2

4

3

90 – 72.5 = 17.5 180 – 72.5 = 107.5

90 – x 180 – x

90 – 2y 180 – 2y

172

2m K

m K x 2m K y

5

6

2 complementary angles are congruent.Find their measures.

x + x = 90

2x = 90 x = 45

450 and 450

2 supplementary angles are congruent.Find their measures.

x + x = 180 2x = 180

x = 90

900 and 900

Name another right angle.7 AFD

F

A E

D

B

C

In the diagram, is a rightangle.AFB

Name the angles.

Two complementary angles.8

AFE and EFD

F

A E

D

B

C

Name the angles.

Two congruent supplementary angles.9

AFB and AFD

F

A E

D

B

C

Name the angles.

Two noncongruent supplementary angles.10

BFE and EFD or

F

A E

D

B

C

Name the angles.

Two noncongruent supplementary angles.10

CFA and AFE

F

A E

D

B

C

Name the angles.

Two acute vertical angles.11

BFC and EFD

F

A E

D

B

C

Name the angles.

Two obtuse vertical angles.12

BFE and CFD

F

A E

D

B

C

Name the angles.

Vertical Angle Th.

Z

Y

XW

VU

TS

In the diagram, bisects andOT��������������

SOU35m UOV

120m YOW

Label completely !

Vertical Angle Th.

35

Vertical Angle Th.

120

6060

6060

60+60+35+x = 180 x = 25

25

25

35

Vertical Angle Th. Now you answer the questions.

O

Z

Y

XW

VU

TS

In the diagram, bisects and

OT��������������

SOU35m UOV

120m YOW

3560

60

606025

25

35

m ZOY

14

13

m ZOW

35

155

O

Z

Y

XW

VU

TS

In the diagram, bisects and

OT��������������

SOU35m UOV

120m YOW

3560

60

606025

25

35

m VOW

16

15

m SOU

25

120

O

Z

Y

XW

VUT

S

In the diagram, bisects and

OT��������������

SOU35m UOV

120m YOW

3560

60

606025

25

35

m TOU

18

17

m ZOT

60

85

O

19(3x-5)

703x -5 = 70

3x = 75

x = 25

divide by 3Vertical Angles

20

(3x+8)

(6x-22)

3x + 8 = 6x - 22

3x + 30 = 6x

30 = 3x

10 = xdivide by 3

Vertical Angles

21

4x

6436

Vertical Angles

4x = 64 + 36

4x = 100

X = 25

Divide by 4

22

are supplements

are supplements

1 2and

3 4and

a] If ,

find .

2 1 4 3

1 3 27m and m 2 4m and m

27 27

180 – 27 = 153

22

are supplements

are supplements

1 2and

3 4and

b] If ,

find .

2 1 4 3

1 3m and m x 2 4m and m

x x

180 – x

22

are supplements

are supplements

1 2and

3 4and

c] If 2 angles are congruent, must their supplements be congruent?

2 1 4 3x x

YES !

y y

23 Given:

Prove: 1 4and 2 3and

Label completely first. 12 3

4

g g

? ?

Statements Reasons

1 2and 2 3and 3 4and 1 4and

Note the flow is better without the given first.

Transitive Prop. Of Equality

Vert. Angles are congruent

Given

Vert. Angles are congruent

24 If and are supplementary,Then find the values of x, and .

A Bm Bm A

2 , 15m A x m B x

Start with a labeled diagram.

A B2x x - 15

2x + x – 15 = 180

3x – 15 = 180

3x = 195

x = 65Divide by 3

A = 2(65)

A = 130

B = 65 - 15

B = 50

25 If and are supplementary,Then find the values of x, and .

A Bm Bm A

16, 2 16m A x m B x Start with a labeled diagram.

A BX + 16 2x - 16

X + 16 +2x– 16 = 180

x = 60

Divide by 3

A = 60 + 16

A = 76

B = 2(60) - 16

B = 120 - 16

3x = 180

B = 104

26 If and are complementary,Then find the values of y, and .

C Dm Dm C

3 5, 2m C y m B y Start with a labeled diagram.

CD

3y+5

2y

3y + 5 + 2y = 90

5y + 5 = 90

5y = 85

y = 17

divide by 5C = 3(17) + 5C = 51 + 5

C = 56

D = 2(17)

D = 34

27 If and are complementary,Then find the values of y, and .

C Dm Dm C

8, 3 2m C y m B y Start with a labeled diagram.

CD

y - 8

3y + 2

y – 8 + 3y + 2 = 90

4y - 6 = 90

4y = 96

y = 24

divide by 4C = 24 - 8

C = 16

D = 3(24) + 2

D = 72 + 2

D = 74

28

Use the information to find an equation and solve.

Find the measure of an angle that is twice as large as its supplement.

2( ) x = 180 – x

x = 180 – 2x

3x = 180

x = 60

180 – 60 = 120

29Use the information to find an equation and solve.

Find the measure of an angle that is half as large as its complement.

x =1

( )2

90 - x

Multipy by 2 to get rid of fractions

22

2x = 90 - x

3x = 90

x = 30

90 – 30 = 60

30

Use the information to find an equation and solve.

The measure of a supplement of an angle is12 more than twice the measure of the angle.

180 – x = 12 + 2x

180 = 12 + 3x

168 = 3x

56 = x

180 – 56 = 124

31

Use the information to find an equation and solve.

A supplement of an angle is six times as large as the complement of the angle.

180 – x = 6( )90 - x

180 – x = 540 – 6x

180 + 5x = 540

5x = 360

x = 72

Supplement

180 – 72 = 108

Complement

90 – 72 = 18

32 Find the values of x and y.

x

(2y – 17)

(3x – 8)

x + 3x – 8 = 180

4x – 8 = 180

4x = 188

x = 47

47

47 + 2y – 17 = 180

2y + 30 = 180

2y = 150

y = 75

33 Find the values of x and y.

50

x3x - y

2x - 16 2x – 16 = 50

2x = 66

x = 33

33

33 = 3(33) - y

33 = 99 - y

- 66 = - y

66 = y

C’est fini.

Good day and good luck.