**Learning Goal**

By the end of the lesson I will be able to . . . describe the properties and relationships of the exterior angles of triangles

What if you knew that two of the exterior angles of a triangle measured

### Watch This

CK-12 Exterior Angles Theorems

James Sousa: Introduction to the Exterior Angles of a Triangle

Then watch this video.

James Sousa: Proof that the Sum of the Exterior Angles of a Triangle is 360 Degrees

Finally, watch this video.

James Sousa: Proof of the Exterior Angles Theorem

### Guidance

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 counter-clockwise.

Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to

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

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.)

#### Example A

Find the measure of

Notice that

Set up an equation to solve for the missing angle.

#### Example B

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

We know that

Similarly,

We also know that the three interior angles must add up to

#### Example C

What is the value of

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

CK-12 Exterior Angles Theorems

### Guided Practice

1. Find

2. Two interior angles of a triangle are

3. Find the value of

**Answers:**

1. Using the Exterior Angle Theorem

If you forget the Exterior Angle Theorem, you can do this problem just like Example C.

2. Remember that every interior angle forms a linear pair (adds up to

So, the measures of the three exterior angles are

3. Set up an equation using the Exterior Angle Theorem.

Substitute in

### Practice

Determine

Use the following picture for the next three problems:

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

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