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

### Exterior Angles in Convex Polygons

Recall that an **exterior angle** is an angle on the outside of a polygon and 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. It does not matter which set you use because one set is just the vertical angles of the other, making the measurement equal. In the picture above, the color-matched angles are vertical angles and congruent. The **Exterior Angle Sum Theorem** stated that the exterior angles of a triangle add up to \begin{align*}360^\circ\end{align*}. Let’s extend this theorem to all polygons.

*Watch the second half of this video.*

#### Investigation: Exterior Angle Tear-Up

Tools Needed: pencil, paper, colored pencils, scissors

- Draw a hexagon like the hexagons above. Color in the exterior angles as well.
- Cut out each exterior angle and label them 1-6.
- Fit the six angles together by putting their vertices together. What happens?

The angles all fit around a point, meaning that the exterior angles of a hexagon add up to \begin{align*}360^\circ\end{align*}, just like a triangle. We can say this is true for all polygons.

**Exterior Angle Sum Theorem:** The sum of the exterior angles of any polygon is \begin{align*}360^\circ\end{align*}.

**Proof of the Exterior Angle Sum Theorem:**

Given: Any \begin{align*}n-\end{align*}gon with \begin{align*}n\end{align*} sides, \begin{align*}n\end{align*} interior angles and \begin{align*}n\end{align*} exterior angles.

Prove: \begin{align*}n\end{align*} exterior angles add up to \begin{align*}360^\circ\end{align*}

NOTE: The interior angles are \begin{align*}x_1, x_2, \ldots x_n\end{align*}.

The exterior angles are \begin{align*}y_1, y_2, \ldots y_n\end{align*}.

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

1. Any \begin{align*}n-\end{align*}gon with \begin{align*}n\end{align*} sides, \begin{align*}n\end{align*} interior angles and \begin{align*}n\end{align*} exterior angles. | Given |

2. \begin{align*}x_n^\circ\end{align*} and \begin{align*}y_n^\circ\end{align*} are a linear pair |
Definition of a linear pair |

3. \begin{align*}x_n^\circ\end{align*} and \begin{align*}y_n^\circ\end{align*} are supplementary |
Linear Pair Postulate |

4. \begin{align*}x_n^\circ+ y_n^\circ=180^\circ\end{align*} | Definition of supplementary angles |

5. \begin{align*}(x_1^\circ+x_2^\circ+\ldots+x_n^\circ)+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)=180^\circ n\end{align*} | Sum of all interior and exterior angles in an \begin{align*}n-\end{align*}gon |

6. \begin{align*}(n-2)180^\circ=(x_1^\circ+ x_2^\circ+\ldots+x_n^\circ)\end{align*} | Polygon Sum Formula |

7. \begin{align*}180^\circ n=(n-2)180^\circ+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Substitution PoE |

8. \begin{align*}180^\circ n=180^\circ n-360^\circ+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Distributive PoE |

9. \begin{align*}360^\circ=(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Subtraction PoE |

#### Solving for Unknown Angle Measurments

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

\begin{align*}y\end{align*} is an exterior angle, as well as all the other given angle measures. Exterior angles add up to \begin{align*}360^\circ\end{align*}, so set up an equation.

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

#### Measuring Exterior Angles

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

Because the polygon is regular, each interior angle is equal. This also means that all the exterior angles are equal. The exterior angles add up to \begin{align*}360^\circ\end{align*}, so each angle is \begin{align*}\frac{360^\circ}{7} \approx 51.43^\circ\end{align*}.

#### Calculating the Sum of Exterior Angles

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*}.

#### Earlier Problem Revisited

The exterior angles of a regular polygon sum to \begin{align*}360^\circ\end{align*}. The measure of each exterior angle in a dodecagon (twelve-sided regular polygon) is \begin{align*}\frac{360^\circ}{12} = 30^\circ\end{align*}.

### Examples

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

#### Example 1

12-gon

For each, divide by \begin{align*}360^\circ\end{align*} and by the given number of sides.

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

#### Example 2

100-gon

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

#### Example 3

36-gon

\begin{align*}10^\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
- 50-gon
- heptagon
- 34-gon
Each interior angle forms a linear pair with an exterior angle. In a regular polygon you can use two different formulas to find the measure of each exterior angle. One way is \begin{align*}\frac{360^\circ}{n}\end{align*} and the other is \begin{align*}180^\circ - \frac{(n-2)180^\circ}{n}\end{align*} (\begin{align*}180^\circ\end{align*} minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.*Challenge*Find the measures of the lettered angles below given that \begin{align*}m \ || \ n\end{align*}.*Angle Puzzle*

### Review (Answers)

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