# 4.18: Exterior Angles and Theorems

- Page ID
- 4815

Exterior angles equal the sum of the remote interiors.

### Exterior Angles

An **Exterior Angle** is the angle formed by one side of a polygon and the extension of the adjacent side.

In all polygons, there are ** two** sets of exterior angles, one that goes around clockwise and the other goes around counterclockwise.

Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to \(180^{\circ}\).

\(m\angle 1+m\angle 2=180^{\circ} \)

There are two important theorems to know involving exterior angles: the **Exterior Angle Sum Theorem** and the Exterior Angle Theorem.

The **Exterior Angle Sum Theorem** states that the exterior angles of any polygon will always add up to \(360^{\circ}\).

\(m\angle 1+m\angle 2+m\angle 3=360^{\circ}\)

\(m\angle 4+m\angle 5+m\angle 6=360^{\circ}\).

The **Exterior Angle Theorem** states that an exterior angle of a triangle is equal to the sum of its **remote interior angles**. (**Remote Interior Angles** are the two interior angles in a triangle that are not adjacent to the indicated exterior angle.)

\(m\angle A+m\angle B=m\angle ACD\)

What if you knew that two of the exterior angles of a triangle measured \(130^{\circ}\)? How could you find the measure of the third exterior angle?

Example \(\PageIndex{1}\)

Two interior angles of a triangle are \(40^{\circ}\) and \(73^{\circ}\). What are the measures of the three exterior angles of the triangle?

**Solution**

Remember that every interior angle forms a linear pair (adds up to \(180^{\circ}\)) with an exterior angle. So, since one of the interior angles is \(40^{\circ}\) that means that one of the exterior angles is \(140^{\circ}\) (because \(40+140=180\)). Similarly, since another one of the interior angles is \(73^{\circ}\), one of the exterior angles must be \(107^{\circ}\). The third interior angle is not given to us, but we could figure it out using the **Triangle Sum Theorem**. We can also use the Exterior Angle Sum Theorem. If two of the exterior angles are \(140^{\circ}\) and \(107^{\circ}\), then the third Exterior Angle must be \(113^{\circ}\) since \(140+107+113=360\).

So, the measures of the three exterior angles are 140, 107 and 113.

Example \(\PageIndex{2}\)

Find the value of \(x\) and the measure of each angle.

**Solution**

Set up an equation using the Exterior Angle Theorem.

\(\begin{align*} \underbrace{(4x+2)^{\circ}+(2x−9)^{\circ}}_\text{remote interior angles}&=\underbrace{(5x+13)^{\circ}}_\text{exterior angle} \\ (6x−7)^{\circ}&=(5x+13)^{\circ} \\ x&=20 \end{align*}\)

Substitute in 20 for \(x\) to find each angle.

\([4(20)+2]^{\circ}=82^{\circ}[2(20)−9]^{\circ}=31^{\circ} \qquad Exterior \:angle:\: [5(20)+13]^{\circ}=113^{\circ}\)

Example \(\PageIndex{3}\)

Find the measure of \(\angle RQS\).

**Solution**

Notice that \(112^{\circ}\) is an exterior angle of \(\Delta RQS\) and is supplementary to \(\angle RQS\).

Set up an equation to solve for the missing angle.

\(\begin{align*}112^{\circ}+m\angle RQS &=180^{\circ} \\ m\angle RQS&=68^{\circ}\end{align*}\)

Example \(\PageIndex{4}\)

Find the measures of the numbered interior and exterior angles in the triangle.

**Solution**

We know that \(m\angle 1+92^{\circ}=180^{\circ}\) because they form a linear pair. So, m\angle 1=88^{\circ}\).

Similarly, \(m\angle 2+123^{\circ}=180^{\circ}\) because they form a linear pair. So, m\angle 2=57^{\circ}\).

We also know that the three interior angles must add up to 180^{\circ}\) by the Triangle Sum Theorem.

\(\begin{align*} m\angle 1+m\angle 2+m\angle 3&=180^{\circ} \qquad by\: the \:Triangle \:Sum \:Theorem. \\ 88^{\circ}+57^{\circ}+m\angle 3&=180 \\ m\angle 3&=35^{\circ}\end{align*}\)

Lastly, \(m\angle 3+m\angle 4=180^{\circ} \qquad because\: they\: form \:a \:linear \:pair.\)

\(\begin{align*} 35^{\circ}+m\angle 4&=180^{\circ} \\ m\angle 4&=145^{\circ}\end{align*}\)

Example \(\PageIndex{5}\)

What is the value of \(p\) in the triangle below?

**Solution**

First, we need to find the missing exterior angle, which we will call \(x\). Set up an equation using the Exterior Angle Sum Theorem.

\(\begin{align*} 130^{\circ}+110^{\circ}+x&=360^{\circ} \\ x&=360^{\circ}−130^{\circ}−110^{\circ} \\ x&=120^{\circ}\end{align*} \)

\(x\) and \(p\) add up to \(180^{\circ}\) because they are a linear pair.

\(\begin{align*} x+p&=180^{\circ} \\ 120^{\circ}+p&=180^{\circ} \\ p&=60^{\circ}\end{align*}\)

### Review

Determine \(m\angle 1\).

Use the following picture for the next three problems:

- What is \(m\angle 1+m\angle 2+m\angle 3\)?
- What is \(m\angle 4+m\angle 5+m\angle 6\)?
- What is \(m\angle 7+m\angle 8+m\angle 9\)?

Solve for \(x\).

## Resources

## Vocabulary

Term | Definition |
---|---|

Exterior angles |
An exterior angle is the angle formed by one side of a polygon and the extension of the adjacent side. |

interior angles |
The angles on the inside of a polygon. |

remote interior angles |
The remote interior angles (of a triangle) are the two interior angles that are not adjacent to the indicated exterior angle. |

Triangle Sum Theorem |
The states that the three interior angles of any triangle will always add up to \(180^{\circ}\).Triangle Sum Theorem |

Exterior Angle Sum Theorem |
Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to 360 degrees. |

## Additional Resources

Interactive Element

Video: Exterior Angles Theorems Examples - Basic

Activities: Exterior Angles Theorems Discussion Questions

Study Aids: Triangle Relationships Study Guide

Practice: Exterior Angles and Theorems

Real World: Exterior Angles Theorem