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 . 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 , 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 .

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

Given: Any gon with sides, interior angles and exterior angles.

Prove: exterior angles add up to

NOTE: The interior angles are .

The exterior angles are .

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

1. Any gon with sides, interior angles and exterior angles. | Given |

2. and are a linear pair |
Definition of a linear pair |

3. and are supplementary |
Linear Pair Postulate |

4. | Definition of supplementary angles |

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

6. | Polygon Sum Formula |

7. | Substitution PoE |

8. | Distributive PoE |

9. | Subtraction PoE |

#### Solving for Unknown Angle Measurments

What is ?

is an exterior angle, as well as all the other given angle measures. Exterior angles add up to , so set up an equation.

#### 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 , so each angle is .

#### 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 . The measure of each exterior angle in a dodecagon (twelve-sided regular polygon) is .

### Examples

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

#### Example 1

12-gon

For each, divide by and by the given number of sides.

#### 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 and What is ?

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 and the other is ( minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.*Challenge*Find the measures of the lettered angles below given that .*Angle Puzzle*

### Review (Answers)

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