Angle Pair Relationships

L.T. I can identify special angle pairs and use their relationships to find angle

measure.

A. Vertical Angles

Previously, you learned that two angles are adjacent if they share a common vertex and side but have no common interior points. In this lesson, you will study other relationships between pairs of angles.

1 and 3 are vertical angles.

2 and 4 are vertical angles.

14

3

2

Two angles are vertical angles if their sides form two pairs of opposite rays.

Vertical Angle Pairs are CONGRUENT

B. Linear Pairs

5 and 6 are a linear pair.

5 6

Two adjacent angles are a linear pair if the form a straight line.

Linear Angle Pairs add up to 180°.

30° 150°

Finding Angle Measures

In the stair railing shown, 6 has a measure of 130˚. Find the measures of the other three angles.

SOLUTION

6 and 8 are vertical angles. So, they are congruent and have the same measure.

m 8 = m 6 = 130˚

5

67

8130°

130°

6 and 7 are a linear pair. So, the sum of their measures is 180˚.

m6 + m7 = 180˚

130˚ + m7 = 180˚

m7 = 50˚

5

67

8

7 and 5 are vertical angles. So, they are congruent and have the same measure.

5

67

8

50°

50°

130°130°

All 4 angles together equal 360°

Definition: Two angles are supplementary if the sum of their measures is 180 degrees. Each angle is the supplement of the other.

1 2

20160

These are supplements of each other because their angles add up to 180.

C. Supplementary Angles

x

Example 1 Find the value of x.

x + = 18020

x = 160

20

x

Example 2 Find the value of x.

65

x + = 18065x = 115

Example 3 Find the value of x.

(7x 10) 3x

(7x + 10) + 3x = 180 10x + 10 = 180

10x = 170

x = 17

Definition: Two angles are complementary if the sum of their measures is 90 degrees. Each angle is the complement of the other.

12

3060

These are complements of each other because their angles add up to be 90.

D. Complementary Angles

Example 4 Find the value of x.

x

15x + = 9015

x = 75

Example 5 Find the value of x.

(4x + 3)

(x - 8)

(4x + 3) + (x - 8) = 90

x = 19

5x - 5 = 905x = 95

Definition: An angle bisector is a ray that divides an angle into two congruent angles. It cuts the angle in half.

E. Angle Bisector