1.5 Exploring

Angle Pairs

Adjacent Angles

Two angles that share a common vertex and side but no common interior points.

1 and ADC are not adjacent.

Adjacent Angles( a common side )

Non-Adjacent Angles

22°

36°

21

D

B

C

A

4

3

Definition:

Examples:

1 and 2 are adjacent.

3 and 4 are not adjacent.

Complementary Angles

A pair of angles whose sum is 90˚Definition:

Examples:

Adjacent Angles( a common side )

21

Q

AB

C 1

2

Q

R

AB

F

G

Non-Adjacent Angles

m1 = 40°m2 = 50°

Supplementary Angles

A pair of angles whose sum is 180˚Definition:

Examples:

Adjacent supplementary angles are also called “Linear Pair.”

Non-Adjacent Angles

2 1

A Q

B

C

1

2

A QR

BF

Gm2 = 140°

m1 = 40°

Examples

< 1 and < 2 are complementary angles. Given m < 1, find m < 2.

a. 521m b. 191m

< 3 and < 4 are supplementary angles. Given m < 3, find m < 4.

b. a. 383m 1473m

382m 712m

334m 1424m

Examples

< A and < B are complementary angles. Find m < A & m < B.

107

45

xBm

xAm 5 𝑥+4+7 𝑥−10=9012𝑥−6=90

12𝑥=96𝑥=8

𝑚<𝐴=5 (8 )+4

𝑚<𝐴=40+4

𝑚<𝐴=44

𝑚<𝐵=7 (8 )−10

𝑚<𝐵=56−10

𝑚<𝐵=46

Examples

< C and < D are supplementary angles. Find m < C & m < D.

1

37

xDm

xCm

7 𝑥−3+𝑥−1=1808 𝑥−4=180

8 𝑥=184𝑥=23

𝑚<𝐶=7 (23 )−3

𝑚<𝐴=161−3

𝑚<𝐴=158

𝑚<𝐵=(23 )−1

𝑚<𝐵=23−1

2

Linear Pair

Two adjacent angles are a linear pair if their non-common sides are opposite rays.

Definition:

The angles in a linear pair are supplementary.

Vertical Angles

A pair of angles whose sides form opposite rays.

Definition:

4

3

2

1A

Q

D

B

C

2 and 4

1 and 3

Vertical angles are non-adjacent angles formed by intersecting lines.

Vertical angles are congruent.

Examples

What are the linear pairs?

What are the vertical angles?

<4 and <5

<1 and <5

Examples< 1 & < 3

< 2 & < 3

< 4 & < 5

< 8 & < 5

< 6 & < 7

< 4 & < 9

< 1 & < 2 & < 3

neither

Linear pair

Vertical angles

Linear pair

Vertical angles

neither

neither

Example: If m4 = 67º, find the measures of all other angles.

3 4 180m m

3 67 180m

3 180 67 113m

3 1 , . 3 1 117 Because and are vertical angles they are equal m m

4

3

2

1

67º

Step 1: Mark the figure with given info.

Step 2: Write an equation.

4 2 , . 4 2 67 Because and arevertical angles they are equal m m

Example: If m1 = 23 º and m2 = 32 º, find the measures of all other angles.

4 23 ( 1 & 4 .)

5 32 ( 2 & 5 .)

m are vertical angles

m are vertical angles

1 2 3 180

23 32 3 180

3 180 55 125

3 6 125

3 & 6 .

m m m

m

m

m m

are vertical angles

6 5

4 3 2

1

Answers:

Example: If m 1 = 44º, m 7 = 77º find the measures of all other angles.

3 90m

1 4 44m m

4 5 90

44 5 90

5 46

m m

m

m

6 7 90

6 65 90

6 25

m m

m

m

7

6 5 4

3

2 1

Answers: