Measuring and Classifying Angles (Examples)

Measuring and classifying angles means finding an angle’s size in degrees and naming its type. An angle less than $90^\circ$ is acute, exactly $90^\circ$ is right, between $90^\circ$ and $180^\circ$ is obtuse, and exactly $180^\circ$ is straight. You measure angles with a protractor.

Before this lesson: You should know Points, Lines and Planes (rays) and the Segment Addition Postulate.

What Is an Angle?

An angle is made of two rays that share an endpoint. The shared endpoint is the vertex. The two rays are the sides of the angle.

Think of a door opening. The hinge is the vertex. The wall and the door are the sides. The wider the door opens, the bigger the angle.

How to name an angle

Use three points, with the vertex in the middle: $\angle ABC$ or $\angle CBA$ .
Use the vertex alone, if only one angle is at that point: $\angle B$ .
Use a number written inside the angle: $\angle 1$ .

The measure of $\angle ABC$ is written $m\angle ABC$ . If it is 40 degrees, write $m\angle ABC = 40^\circ$ . Angles with the same measure are congruent angles, written $\angle A \cong \angle B$ . Matching arcs mark them on a drawing.

Types of Angles

Type	Measure	Picture it
Acute	More than $0^\circ$ and less than $90^\circ$	A slice of pizza
Right	Exactly $90^\circ$	The corner of a sheet of paper
Obtuse	More than $90^\circ$ and less than $180^\circ$	A reclining chair
Straight	Exactly $180^\circ$	A flat line through the vertex

Four angles in a row, each with its vertex marked. An acute angle of 40 degrees. A right angle of 90 degrees, shown with a small square at the vertex. An obtuse angle of 130 degrees. A straight angle of 180 degrees, whose two rays point in opposite directions. — Acute is less than 90°, right is exactly 90°, obtuse is between 90° and 180°, and straight is exactly 180°.

A small square at the vertex means the angle is exactly $90^\circ$ . You don’t need to see a number.

Measuring Angles and the Angle Addition Postulate

A protractor has two scales, from 0 to 180 in each direction. Put its center on the vertex. Line up one side with 0 on one scale. Read the other side on that same scale.

This is the protractor postulate: if the sides of an angle cross the scale at $a$ and $b$ , the angle measures $|a - b|$ degrees. It works like the ruler postulate for segments.

The angle addition postulate says that if point $D$ is in the interior of $\angle ABC$ , then:

m\angle ABD + m\angle DBC = m\angle ABC

It is the same idea as the segment addition postulate: part + part = whole.

Angle ABC with vertex B. A ray BD lies between ray BA and ray BC. The blue angle ABD is labeled 47 degrees and the orange angle DBC is labeled 38 degrees. A green arc around the whole angle ABC is labeled 85 degrees. — Ray BD splits angle ABC into two parts: 47° + 38° = 85°.

Steps to measure and classify an angle

Put the protractor’s center on the vertex.
Line up one side with 0 on a scale.
Read where the other side crosses that scale. (Or subtract the two readings.)
Compare the measure with $90^\circ$ and $180^\circ$ to name the type.

Worked Examples

Example 1: Classify angles

Classify angles that measure $35^\circ$ , $90^\circ$ , $128^\circ$ and $180^\circ$ .

Step 1: $35^\circ$ is less than $90^\circ$ , so it is acute.

Step 2: $90^\circ$ is exactly a right angle.

Step 3: $128^\circ$ is between $90^\circ$ and $180^\circ$ , so it is obtuse.

Step 4: $180^\circ$ is a straight angle.

Example 2: Read a protractor

On a protractor, ray $OA$ crosses the scale at $20^\circ$ and ray $OB$ crosses the same scale at $105^\circ$ . Find $m\angle AOB$ and classify it.

Step 1: Use the protractor postulate. Subtract the readings.

m\angle AOB = |105 - 20|

Step 2: Subtract.

m\angle AOB = 85^\circ

Step 3: Compare with $90^\circ$ . 85 is less than 90.

So $m\angle AOB = 85^\circ$ , an acute angle.

Example 3: Add two angles

Point $D$ is in the interior of $\angle ABC$ . $m\angle ABD = 47^\circ$ and $m\angle DBC = 38^\circ$ . Find $m\angle ABC$ .

Step 1: Write the angle addition postulate.

m\angle ABD + m\angle DBC = m\angle ABC

Step 2: Put in the measures.

47^\circ + 38^\circ = m\angle ABC

Step 3: Add.

85^\circ = m\angle ABC

So $m\angle ABC = 85^\circ$ . The picture above shows this angle.

Example 4: Solve for x

Point $D$ is in the interior of $\angle ABC$ . $m\angle ABD = (3x + 5)^\circ$ , $m\angle DBC = (2x - 10)^\circ$ and $m\angle ABC = 120^\circ$ . Find $x$ and both angle measures.

Step 1: Write the angle addition postulate.

(3x + 5) + (2x - 10) = 120

Step 2: Combine like terms: $3x + 2x = 5x$ and $5 - 10 = -5$ (see combining like terms).

5x - 5 = 120

Step 3: Add 5 to both sides.

5x = 125

Step 4: Divide both sides by 5.

x = 25

Step 5: Find $m\angle ABD$ .

3(25) + 5 = 75 + 5 = 80

Step 6: Find $m\angle DBC$ .

2(25) - 10 = 50 - 10 = 40

Step 7: Check: the parts add up to the whole.

80 + 40 = 120

So $x = 25$ , $m\angle ABD = 80^\circ$ and $m\angle DBC = 40^\circ$ . Both parts are acute, and the whole angle is obtuse.

Example 5: Word problem

Two streets meet at a corner that makes a $140^\circ$ angle. A straight bike path starts at the corner and splits the angle into two parts. One part is $26^\circ$ more than the other. Find both angles.

Step 1: Let the smaller angle be $x$ degrees. The larger angle is $x + 26$ .

Step 2: Write part + part = whole.

x + (x + 26) = 140

Step 3: Combine like terms.

2x + 26 = 140

Step 4: Subtract 26 from both sides.

2x = 114

Step 5: Divide both sides by 2.

x = 57

Step 6: Find the larger angle.

57 + 26 = 83

Step 7: Check: the parts add up to the whole.

57 + 83 = 140

The angles are $57^\circ$ and $83^\circ$ . Both are acute.

Common Mistakes

Mistake	Example of the error	Fix
Reading the wrong scale	Reading $95^\circ$ for an angle that is clearly acute	Start from the 0 that one side lines up with, and use that scale. Check: does the answer match the look of the angle?
Putting the vertex in the wrong place in the name	Calling the angle at $B$ $\angle BAC$	The vertex goes in the middle: $\angle ABC$ .
Using $\angle B$ when several angles share $B$	Writing $\angle B$ when $\angle ABD$ , $\angle DBC$ and $\angle ABC$ all have vertex $B$	Use three letters so the reader knows which angle you mean.
Stopping at $x$	Answering $x = 25$ in Example 4	Put $x$ back in to find each angle: $80^\circ$ and $40^\circ$ .
Calling $90^\circ$ obtuse or acute	Saying a right angle is acute	Exactly $90^\circ$ is right. Acute is less, and obtuse is more.

Angle Bisectors

An angle bisector is a ray that splits an angle into two congruent angles. It does for angles what a midpoint does for segments.

Say $\overrightarrow{BD}$ bisects $\angle ABC$ , with $m\angle ABD = (4x - 6)^\circ$ and $m\angle DBC = (2x + 18)^\circ$ . The two halves are equal.

\begin{aligned} 4x - 6 &= 2x + 18 && \text{the halves are equal} \\ 2x - 6 &= 18 && \text{subtract } 2x \\ 2x &= 24 && \text{add 6} \\ x &= 12 && \text{divide by 2} \end{aligned}

Each half is $4(12) - 6 = 42^\circ$ , so $m\angle ABC = 42^\circ + 42^\circ = 84^\circ$ . To review solving equations like this, see equations with variables on both sides.

Practice Problems

Classify angles that measure $72^\circ$ , $90^\circ$ , $145^\circ$ and $180^\circ$ .
On a protractor, one side of an angle crosses the scale at $30^\circ$ and the other at $115^\circ$ . Find the measure and classify the angle.
On a protractor, one side crosses the scale at $160^\circ$ and the other at $45^\circ$ . Find the measure and classify the angle.
Point $S$ is in the interior of $\angle PQR$ . $m\angle PQS = 28^\circ$ and $m\angle SQR = 54^\circ$ . Find $m\angle PQR$ .
Point $D$ is in the interior of $\angle ABC$ . $m\angle ABC = 150^\circ$ and $m\angle ABD = 65^\circ$ . Find $m\angle DBC$ .
Point $D$ is in the interior of $\angle ABC$ . $m\angle ABD = (2x + 7)^\circ$ , $m\angle DBC = (3x - 2)^\circ$ and $m\angle ABC = 95^\circ$ . Find both angles.
$\overrightarrow{BD}$ bisects $\angle ABC$ . $m\angle ABD = (5x - 8)^\circ$ and $m\angle DBC = (3x + 12)^\circ$ . Find $m\angle ABC$ .
$\angle ABC$ is a straight angle, and $D$ is in its interior. $m\angle ABD = (4x + 10)^\circ$ and $m\angle DBC = (2x + 20)^\circ$ . Find both angles and classify them.
The hands of a clock turn $30^\circ$ for each hour mark. What angle do the hands make at 5:00? Classify it.
A right angle is split into two angles. The larger one is twice the smaller one. Find both angles.

Show answers

72° is less than 90°, so it is acute. 90° is right. 145° is between 90° and 180°, so it is obtuse. 180° is straight.
Subtract the readings: $|115 - 30| = 85$ . The angle is $85^\circ$ , so it is acute.
Subtract the readings: $|45 - 160| = |-115| = 115$ . The angle is $115^\circ$ , so it is obtuse.
Write the postulate: $m\angle PQS + m\angle SQR = m\angle PQR$ . Put in the measures: $28 + 54 = m\angle PQR$ . Add: $m\angle PQR = 82^\circ$ . So $m\angle PQR = 82^\circ$ .
Write the postulate: $65 + m\angle DBC = 150$ . Subtract 65 from both sides: $m\angle DBC = 150 - 65$ . Subtract: $m\angle DBC = 85^\circ$ . So $m\angle DBC = 85^\circ$ .
Write the postulate: $(2x + 7) + (3x - 2) = 95$ . Combine like terms: $5x + 5 = 95$ . Subtract 5 from both sides: $5x = 90$ . Divide both sides by 5: $x = 18$ . Find the first angle: $2(18) + 7 = 43$ . Find the second angle: $3(18) - 2 = 52$ . Check: $43 + 52 = 95$ . So $m\angle ABD = 43^\circ$ and $m\angle DBC = 52^\circ$ .
The halves are equal: $5x - 8 = 3x + 12$ . Subtract 3x from both sides: $2x - 8 = 12$ . Add 8 to both sides: $2x = 20$ . Divide both sides by 2: $x = 10$ . Find each half: $5(10) - 8 = 42$ . Add the halves: $42 + 42 = 84$ . So $m\angle ABC = 84^\circ$ .
A straight angle is 180°: $(4x + 10) + (2x + 20) = 180$ . Combine like terms: $6x + 30 = 180$ . Subtract 30 from both sides: $6x = 150$ . Divide both sides by 6: $x = 25$ . Find the first angle: $4(25) + 10 = 110$ . Find the second angle: $2(25) + 20 = 70$ . Check: $110 + 70 = 180$ . So $m\angle ABD = 110^\circ$ (obtuse) and $m\angle DBC = 70^\circ$ (acute).
At 5:00, the hands are 5 hour marks apart: $5 \cdot 30 = 150$ . The angle is $150^\circ$ , so it is obtuse.
Let the smaller angle be x. The larger is 2x. Write part + part = whole: $x + 2x = 90$ . Combine like terms: $3x = 90$ . Divide both sides by 3: $x = 30$ . Find the larger angle: $2(30) = 60$ . The angles are $30^\circ$ and $60^\circ$ .

FAQ

What are the four types of angles?

Acute (less than $90^\circ$ ), right (exactly $90^\circ$ ), obtuse (between $90^\circ$ and $180^\circ$ ) and straight (exactly $180^\circ$ ). Angles more than $180^\circ$ are called reflex angles.

How do you know which protractor scale to read?

Use the scale that starts at 0 on the side you lined up. A quick check helps too: an angle that looks small should get a measure under 90.

What is the angle addition postulate?

If point $D$ is inside $\angle ABC$ , then $m\angle ABD + m\angle DBC = m\angle ABC$ . The two smaller angles add up to the whole angle.

What is the difference between equal and congruent angles?

Congruent describes the angles: $\angle A \cong \angle B$ . Equal describes their measures: $m\angle A = m\angle B$ . If one is true, so is the other.

Summary

An angle is two rays with a shared endpoint, the vertex. Name it with the vertex in the middle, like $\angle ABC$ .
Acute: less than $90^\circ$ . Right: $90^\circ$ . Obtuse: between $90^\circ$ and $180^\circ$ . Straight: $180^\circ$ .
Measure with a protractor, or subtract the two readings on the same scale.
Angle addition postulate: $m\angle ABD + m\angle DBC = m\angle ABC$ .
An angle bisector splits an angle into two congruent angles.

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Next lesson: Angle Pairs

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