### Exterior Angle Sum Theorem

An exterior angle is an angle that is formed by extending a side of the polygon.

As you can see, there are two sets of exterior angles for any vertex on a polygon, one going around clockwise (\begin{align*}1^{st}\end{align*} hexagon), and the other going around counter-clockwise (\begin{align*}2^{nd}\end{align*} hexagon). The angles with the same colors are vertical and congruent.

The **Exterior Angle Sum Theorem** states that the sum of the exterior angles of ANY convex polygon is \begin{align*}360^\circ\end{align*}. If the polygon is regular with \begin{align*}n\end{align*} sides, this means that each exterior angle is \begin{align*}\frac{360}{n}^\circ\end{align*}.

What if you were given a seven-sided regular polygon? How could you determine the measure of each of its exterior angles?

### Examples

#### Example 1

What is the measure of each exterior angle of a regular 12-gon?

Divide \begin{align*}360^\circ\end{align*} by the given number of sides.

\begin{align*}30^\circ\end{align*}

#### Example 2

What is the measure of each exterior angle of a regular 100-gon?

Divide \begin{align*}360^\circ\end{align*} by the given number of sides.

\begin{align*}3.6^\circ\end{align*}

#### Example 3

What is \begin{align*}y\end{align*}?

\begin{align*}y\end{align*} is an exterior angle and all the given angles add up to \begin{align*}360^\circ\end{align*}. Set up an equation.

\begin{align*}70^\circ + 60^\circ + 65^\circ + 40^\circ + y & = 360^\circ\\ y & = 125^\circ\end{align*}

#### Example 4

What is the measure of each exterior angle of a regular heptagon?

Because the polygon is regular, the interior angles are equal. It also means the exterior angles are equal. \begin{align*}\frac{360^\circ}{7} \approx 51.43^\circ\end{align*}

#### Example 5

What is the sum of the exterior angles in a regular 15-gon?

The sum of the exterior angles in any convex polygon, including a regular 15-gon, is \begin{align*}360^\circ\end{align*}.

### Review

- What is the measure of each exterior angle of a regular decagon?
- What is the measure of each exterior angle of a regular 30-gon?
- What is the sum of the exterior angles of a regular 27-gon?

Find the measure of the missing variables:

- The exterior angles of a quadrilateral are \begin{align*}x^\circ, 2x^\circ, 3x^\circ,\end{align*} and \begin{align*}4x^\circ.\end{align*} What is \begin{align*}x\end{align*}?

Find the measure of each exterior angle for each regular polygon below:

- octagon
- nonagon
- triangle
- pentagon

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.2.