Measure and Classify Angles

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, 

1. ∠DRY 
2. ∠YRD 
3. ∠R 

Example 2: 

Name this angle in four different ways: 

The above angle can be named as, 

1. ∠WET 
2. ∠TEW 
3. ∠E 
4. ∠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

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

2. Name three different angles in the diagram given below.

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

a) m∠W = 180             b) m∠X = 30  

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

a) m∠Y = 90                 b) m∠Z = 95

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

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

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

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

9. Find the indicated angle measure

10. Given m∠WXZ = 80 , find m∠YXZ,

11. 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