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

*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

**Exterior Angle Sum Theorem:** The sum of the exterior angles of any polygon is

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

Given: Any

Prove:

NOTE: The interior angles are

The exterior angles are

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

1. Any |
Given |

2. and |
Definition of a linear pair |

3. and |
Linear Pair Postulate |

4. |
Definition of supplementary angles |

5. |
Sum of all interior and exterior angles in an |

6. |
Polygon Sum Formula |

7. |
Substitution PoE |

8. |
Distributive PoE |

9. |
Subtraction PoE |

#### Solving for Unknown Angle Measurments

What is

#### 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

#### 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

#### Earlier Problem Revisited

The exterior angles of a regular polygon sum to

### Examples

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

#### Example 1

12-gon

For each, divide by

#### Example 2

100-gon

#### Example 3

36-gon

### 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
x∘,2x∘,3x∘, and4x∘. What isx ?

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*Challenge*360∘n and the other is180∘−(n−2)180∘n (180∘ minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.Find the measures of the lettered angles below given that*Angle Puzzle*m || n .

### Review (Answers)

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