### Exterior Angle Theorems

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 going around the polygon clockwise and the other goes around the polygon counterclockwise. By the definition, the interior angle and its adjacent exterior angle form a linear pair.

The **Exterior Angle Sum Theorem** states that each set of exterior angles of a polygon add up to

**Remote interior angles** are the two angles in a triangle that are not adjacent to the indicated exterior angle.

The **Exterior Angle Theorem** states that the sum of the remote interior angles is equal to the non-adjacent exterior angle. From the picture above, this means that

Given:

Prove:

Statement |
Reason |
---|---|

1. |
Given |

2. |
Triangle Sum Theorem |

3. |
Linear Pair Postulate |

4. |
Transitive PoE |

5. |
Subtraction PoE |

#### Measuring Angles in a Triangle

1. Find the measure of

If we draw both sets of exterior angles on the same triangle, we have the following figure:

Notice, at each vertex, the exterior angles are also vertical angles, therefore they are congruent.

2. Find the measure of the numbered interior and exterior angles in the triangle.

3. What is the value of

First, we need to find the missing exterior angle, we will call it

#### Using the Exterior Angle Sum Theorem

The third exterior angle of the triangle below is

By the Exterior Angle Sum Theorem:

### Examples

#### Example 1

Find

#### Example 2

Find

Using the Exterior Angle Theorem,

#### Example 3

Find the value of

Set up an equation using the Exterior Angle Theorem.

Substituting

### Explore More

Determine \begin{align*}m\angle{1}\end{align*}.

Use the following picture for the next three problems:

- What is \begin{align*}m\angle{1}+m\angle{2}+m\angle{3}\end{align*}?
- What is \begin{align*}m\angle{4}+m\angle{5}+m\angle{6}\end{align*}?
- What is \begin{align*}m\angle{7}+m\angle{8}+m\angle{9}\end{align*}?

Solve for \begin{align*}x\end{align*}.

- Suppose the measures of the three angles of a triangle are x, y, and z. Explain why \begin{align*}x+y+z=180\end{align*}.
- Suppose the measures of the three angles of a triangle are x, y, and z. Explain why the expression \begin{align*}(180-x)+(180-y)+(180-z)\end{align*} represents the sum of the exterior angles of the triangle.
- Use your answers to the previous two problems to help justify why the sum of the exterior angles of a triangle is 360 degrees. Hint: Use algebra to show that \begin{align*}(180-x)+(180-y)+(180-z)\end{align*} must equal 360 if \begin{align*}x+y+z=180\end{align*}.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.2.