## Key Concepts

- Identify complements and supplements
- Find measures of a complement and a supplement
- Find angle measures
- Identify angle pairs
- Find angle measures in a linear pair

## Describe Angle Pair Relationships

### Introduction

In this chapter, we will learn to identify complementary angles and supplementary angles, find the measures of a complement and a supplement. Find the angle measures, identify angle pairs, find angle measures in a linear pair, and learn about angle pair relationships.

### Complementary angles

Two angles are complementary angles if the sum of their measures is 90^{o}.

Each angle is the complement of the other.

### Supplementary angles

Two angles are supplementary angles if the sum of their measures is 180o.

Each angle is the supplement of the other.

#### Identify complements and supplements

**Example 1:**

In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

**Solution:**

- Because 32
^{°}+ 58^{°}= 90^{°}, ∠BAC and ∠RST are complementary angles. - Because 122° + 58° = 180°, ∠CAD and ∠RST are supplementary angles.
- Because∠BAC and ∠CAD share a common vertex and side, they are adjacent.

#### Find measures of a complement and a supplement

**Example 1:**

Given that ∠1 is a complement of ∠2 and *m*∠1 = 68°, find *m*∠2*.*

**Solution:**

You can draw a diagram with complementary adjacent angles to illustrate the relationship.

**Example 2:**

Given that ∠3 is a supplement of ∠4 and m∠4 = 56°, find m∠3.

**Solution:**

You can draw a diagram with supplementary adjacent angles to illustrate the relationship.

#### Find angle measures

**Example:**

**Sports: **When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find *m*

∠BCE and *m*∠ECD.

**Solution:**

**Step 1: Use the fact that the sum of the measures of supplementary angles is 180°.**

**Step 2: ****Evaluate: the original expressions when ***x ***= 34.**

*m*∠BCE = (4*x *+ 8)° = (34 + 8)° = 144°

*m*∠ECD = (*x *+ 2)° = (34 + 2)° = 36°

**The angle measures are 144° and 36°.**

### Angle pairs

- Two adjacent angles are a
**linear pair**if their non-common sides are opposite rays. The angles in a linear pair are always supplementary.

∠1 and ∠2 are a linear pair.

Let us see another example,

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

∠3 and ∠6 are vertical angles.

∠4 and ∠5 are vertical angles.

Let us see another example,

∠1=∠3

∠2=∠4

#### Identify angle pairs

**Example 1:**

Identify all of the linear pairs and all of the vertical angles in the figure below.

**Solution:**

- To find vertical angles, look for angles formed by intersecting lines.
- ∠1∠1 and ∠5∠5 are vertical angles.
- To find linear pairs, look for adjacent angles whose non-common sides are opposite rays.
- ∠1∠1 and ∠4∠4 are a linear pair. ∠4∠4 and ∠5∠5 are also a linear pair.

#### Find angle measures in a linear pair

**Example:**

Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle.

**Solution:**

Let x° be the measure of one angle. The measure of the other angle is 5x°. Then use the fact that the angles of a linear pair are supplementary to write an equation.

## Exercise

- In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

- In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

- Given that Ðl is a complement of Ð2 and mÐl = 73°, find mÐ2.
- Given that Ðl is a complement of Ð2 and mÐ2 = 57°, find mÐ1.
- Given that Ð3 is a supplement of Ð4 and mÐ4 = 37°, find mÐ3.
- Given that Ð3 is a supplement of Ð4 and mÐ4 = 41°, find mÐ3.
- Find the values of x and y.

- The basketball pole forms a pair of supplementary angles with the ground. Find m and m .

- Identify all of the linear pairs and all of the vertical angles in the figure below.

- Identify all of the linear pairs and all of the vertical angles in the figure.

### What have we learned

- To identify complementary angles and supplementary angles
- To find the measures of a complement and a supplement
- To find the angle measures
- To identify angle pairs
- To find angle measures in a linear pair

### Concept Map

## Frequently asked questions

### 1. What are the different types of angle pairs?

**Ans: **The different types of angle pairs are Complimentary angle pairs, Supplementary angle pairs, linear angle pairs, vertical angle pairs and adjacent angle pairs.

### 2. How do you identify an angle pair?

**Ans:** You can identify angle pairs by the angle measurements or the angle placement.

### 3. How are supplementary angles different from complimentary angles?

**Ans:** Two angles are supplementary angles if the sum of their angle measures is 180°. Meanwhile its complementary angles if the sum of their angle measures is 90°.

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