## Key Concepts

- Name angles
- Measure and classify angles
- Find angle measures
- Identify congruent angles
- Double an angle measure

### Introduction

In this chapter, we will learn to define, classify, draw, name, and measure angles, use the protractor and angle addition postulates and double an angle measure.

### Angle

An angle consists of two different rays (sides) that share the same endpoint (vertex).

Angle ABC, ∠ABC, or ∠B

**Sides:**

Sides are the rays.

**Vertex:**

It is the point where the two rays meet.

**Example:**

In the above example, the angle with sides AB and AC- can be named or . Point A is the vertex of the angle.

### Name angles

We can follow the rules below while naming angles:

- Use three capital letters – Vertex in the middle
- Can use one capital letter if it is the vertex and it is obvious which angle you are referring to
- Can use the number located inside the angle

**Example 1:**

Name this angle in three different ways:

The above angle can be named as,

- ∠DRY
- ∠YRD
- ∠R

**Example 2:**

Name this angle in four different ways:

The above angle can be named as,

- ∠WET
- ∠TEW
- ∠E
- ∠1

**Example 3:**

Name the three angles in the diagram.

The three angles are,

∠*WXY*, or ∠*YXW*

∠*YXZ*, or ∠*ZXY*

∠*WXZ*, or ∠*ZXW*

Why should you NOT label any of these angles “Angle *X*“?

You should not name any of these angles ∠*X* because all three angles have *X* as their vertex.

### POSTULATE 3 Protractor Postulate

The m∠AOB is equal to the absolute value of the difference between the real numbers for OA and OB m∠AOB = |55° – 180°|

m∠AOB = |–125°|

m∠AOB = 125°

## Measure and classify angles

The angle is measured using a protractor in degrees. It is the smallest amount of rotation about the vertex from one side to the other.

**Example:**

- m∠APB = 60°
- m∠APC = 100°
- m∠BPC = 40°

### Types of angles

**Acute angle**: between 0° and 90°

**Right angle**: exactly 90°°

**Obtuse Angle**: between 90° ° and 180°°

**Straight angle**: exactly 180°°

### Angle Addition Postulate

Smaller angles can be added together to form larger angles if they share a common vertex.

If B is in the interior of ∠AOC, then the m∠AOC is equal to the sum of m∠AOB and m∠BOC.

**m**∠**AOC = m**∠**AOB + m**∠**BOC**

### Find angle measures

**Example:**

Given that m∠LKN = 145° find m∠LKM and m∠MKN.

STEP 1: Write and solve an equation to find the value of x.

STEP 2: Evaluate the given expressions when x = 23.

m∠LKM = (2x + 10)°= (2.23 + 10)° = 56°

m∠MKN = (4x – 3)°= (4.23 – 3)° = 89°

So, m∠LKM = 56°

and m∠MKN = 89°.

### Congruent Angles

Congruent angles have the same angle measure.

**Example:**

Can be marked using the same number of hash marks:

#### Identify congruent angles

In the given picture, identify the angles that are congruent. If m∠DEG = 157° then find m∠GKL?

**Solution:**

There are two pairs of congruent angles:

∠DEF ≅ ∠JKL and ∠DEG ≅ ∠GKL.

Because ∠DEG ≅ ∠GKL, m∠DEG = m∠GKL. So, m∠GKL = 157°.

### Double an angle measure

In the below diagram, YW bisects ∠XYZ, and m∠XYW = 18°. Find m∠XYZ.

**Solution:**

By the Angle Addition Postulate, m∠XYZ = m∠XYW + m∠WYZ. Because YW bisects ∠XYZ, you know that ∠XYW ≅ ∠WYZ.

So, m∠XYW = m∠WYZ, and you can write

m∠XYZ = m∠ XYW + m∠WYZ = 18° + 18° = 36°.

## Exercise

- Write three names for the angle shown. Then name the vertex and sides of the angle.

- Name three different angles in the diagram given below.

- Classify the angle with the given measure as acute, obtuse, right, or straight.

a) m∠W = 180 b) m∠X = 30

- Classify the angle with the given measure as acute, obtuse, right, or straight.

a) m∠Y = 90 b) m∠Z = 95

- Use the diagram to find the angle measure. Then classify the angle.

- ∠BOC =
- ∠AOB =
- ∠DOB =
- ∠DOE =

- Use the diagram to find the angle measure. Then classify the angle.

- ∠AOC =
- ∠BOE =
- ∠EOC =
- ∠COD =

- Fill in the blanks with the help of the below given picture.

- Angles with the same measure are congruent angles. This means that if _____________________, then __________________. You can also say that if _______________________, then ______________________.

- Find the indicated angle measure

- Given m∠WXZ = 80 , find m∠YXZ,

- In the below diagram, bisects ∠XWY, and m∠XWZ = 52 . Find m∠YWZ.

### What have we learned

- To define, classify, draw, name, and measure angles.
- To use the protractor and angle addition postulates.
- To double an angle measure

### Concept Map

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: