Math resources Geometry

Angles in polygons

Angles of a triangle

# Angles of a triangle

Here you will learn about angles of a triangle including what the sum of both interior and exterior angles of a triangle are, how to find missing angles, and how to use this alongside other angle facts to form and solve equations.

Students will first learn about angles of a triangle as a part of geometry in 8 th grade and will extend their knowledge throughout high school geometry.

## What are angles of a triangle?

Angles of a triangle refer to the sum of the angles of a triangle, found at each vertex in a triangle.

A triangle has both interior angles and exterior angles.

### Interior angles of a triangle

Interior angles of a triangle are angles that are formed inside a triangle by its three sides. Each interior angle is formed by two adjacent sides of the triangle.

The sum of the interior angles of a triangle is \bf{180}^{\circ}.

For example,

### Exterior angles of a triangle

The exterior angles of a triangle are angles that are formed on the outside of the triangle when its sides are extended.

The sum of exterior angles of a triangle is \bf360^{\circ}.

The exterior angle is also equal to the sum of the two opposite interior angles. Each exterior angle is supplementary to its adjacent interior angle.

### Other angle facts

Sometimes the problem will involve using other angle facts. Let’s recap some of the other important angle facts:

## Common Core State Standards

How does this relate to 8 th grade math and high school math?

• Grade 8: Geometry (8.G.A.5)
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

• High school: Geometry (HS.G.CO.C.10)
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180^{\circ}; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

## How to find a missing angle of a triangle

In order to find the measure of a missing angle of a triangle:

1. Add up the angles that are given within the triangle.
2. Subtract this total from \bf{180}^{\circ}.

## Angles of a triangle examples

### Example 1: scalene triangle

Find the measure of the unknown angle labeled a in the following triangle:

1. Add up the angles that are given within the triangle.

The angles 57^{\circ} and 79^{\circ} are given. Add these together:

57+79=136^{\circ}

2Subtract this total from \bf{180}^{\circ}.

Subtract 136^{\circ} from 180^{\circ} \text{:}

\begin{aligned}& 180-136=44^{\circ} \\\\ & a=44^{\circ}\end{aligned}

### Example 2: right triangle

Find the measure of the unknown angle labeled b in the following triangle:

Add up the angles that are given within the triangle.

Subtract this total from \bf{180}^{\circ}.

### Example 3: isosceles triangle

Find the measure of the unknown angle labeled c in the following triangle:

When two sides of a triangle are equal, the angles at the ends of those sides will also be equal.

Add up the angles that are given within the triangle.

Subtract this total from \bf{180}^{\circ}.

## How to find a missing angle of a triangle using other angle facts

In order to find the missing angle of a triangle using other angle facts:

1. Use angle facts to identify all possible angles.
2. Calculate the missing angles in the triangle.

### Example 4: using angles at a point

Find the measure of the angle labeled e \text{:}

Use angle facts to identify all possible angles.

Calculate the missing angles in the triangle.

### Example 5: using opposite angles

Find the measure of the angle labeled f \text{:}

Use angle facts to identify all possible angles.

Calculate the missing angles in the triangle.

### Example 6: two different triangles

Find the measure of the angle labeled g \text{:}

Use angle facts to identify all possible angles.

Calculate the missing angles in the triangle.

### Teaching tips for angles of a triangle

• Visual aids, such as diagrams, and hands-on activities allow students to explore the properties of triangles in a different way.

• Allow students to work in pairs or small groups while working through practice problems on worksheets. This encourages discussions about math between peers, which allows for students to share their personal insights and strategies used.

• Provide real-time feedback to students as they are working through practice problems to clarify any misunderstandings as they arise. The use of a triangle calculator can support this, without having to take any additional time from you.

### Easy mistakes to make

• Mixing up the sum of interior angles and exterior angles
Students may use 360^{\circ} instead of 180^{\circ} for the sum of the interior angles of the triangle and vice versa.

• Equal angles in an isosceles triangle
Selecting the wrong angles when identifying the equal angles in an isosceles triangle (particularly a problem when the equal angles are not at the bottom). The angle that is different in an isosceles triangle is the one between the two sides with equal length.

### Angles in a triangle practice questions

1. Find the measure of the angle b in the following triangle:

33^{\circ}

123^{\circ}

57^{\circ}

213^{\circ}
\begin{aligned}& 90+57=147 \\\\ & 180-147=33^{\circ}\end{aligned}

2. Find the measure of the angle c \text{:}

51^{\circ}

258^{\circ}

78^{\circ}

39^{\circ}

This is an isosceles triangle and the two angles at the bottom of the triangle are equal.

\begin{aligned}& 51+51=102 \\\\ & 180-102=78^{\circ}\end{aligned}

3. Find the measure of angle x in the following triangle:

42^{\circ}

69^{\circ}

138^{\circ}

48^{\circ}

This is an isosceles triangle and the two angles on the right are equal.

180-42=138

138 \div 2 = 69^{\circ}

4. What is the size of each angle in an equilateral triangle?

60^{\circ}

90^{\circ}

30^{\circ}

180^{\circ}

An equilateral triangle has three equal sides so

180 \div 3 = 60^{\circ}

5. Find the measure of the angle labeled w in the following triangle:

24^{\circ}

156^{\circ}

48^{\circ}

78^{\circ}

The angle opposite 24^{\circ} is also 24^{\circ} since vertically opposite angles are equal. The triangle is an isosceles triangle and the two angles on the left are the same size.

180-24=156

156 \div 2 = 78^{\circ}

6. Find the measure of the angle labeled v \text{:}

51^{\circ}

20^{\circ}

129^{\circ}

31^{\circ}

Let’s find the missing angle in the triangle on the left first,

\begin{aligned}& 90+39=129 \\\\ & 180-129=51^{\circ}\end{aligned}

Then use the fact that angles on a straight line add up to 180^{\circ} to find the unlabeled angle in the right hand triangle.

180-51=129^{\circ}

Now, find angle v \text{:}

\begin{aligned}& 129+31=160^{\circ} \\\\ & 180-160=20^{\circ}\end{aligned}

## Angles of a triangle FAQs

Do all triangles have the same interior angle measures?

No, the interior angles within a triangle can vary depending on the triangle. However, all of the measures of angles within a triangle will always equal 180 degrees.

What is the relationship between interior and exterior angles of a triangle?

Each exterior angle of a triangle is supplementary to its adjacent interior angle. The sum of each exterior angle and its adjacent interior angle will equal 180 degrees.

What is the Law of Sines?

The Law of Sines is a principle in trigonometry that relates the side lengths of a triangle to the sines of its angles. It states that in any triangle, the ration of the length of a side to the sine of its opposite angle is constant for all three sides. It can be expressed as:
\cfrac{a}{\sin (A)}=\cfrac{b}{\sin (B)}=\cfrac{c}{\sin (C)}.

## The next lessons are

• Congruence and similarity
• Transformations
• Mathematical proofs
• Trigonometry

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