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# 6.2: Exterior Angles in Convex Polygons

Difficulty Level: At Grade Created by: CK-12
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Practice Exterior Angles in Convex Polygons

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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 360\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

1. Draw a hexagon like the hexagons above. Color in the exterior angles as well.
2. Cut out each exterior angle and label them 1-6.
3. 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 360\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 360\begin{align*}360^\circ\end{align*}.

Proof of the Exterior Angle Sum Theorem:

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

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

NOTE: The interior angles are x1,x2,xn\begin{align*}x_1, x_2, \ldots x_n\end{align*}.

The exterior angles are y1,y2,yn\begin{align*}y_1, y_2, \ldots y_n\end{align*}.

Statement Reason
1. Any n\begin{align*}n-\end{align*}gon with n\begin{align*}n\end{align*} sides, n\begin{align*}n\end{align*} interior angles and n\begin{align*}n\end{align*} exterior angles. Given
2. xn\begin{align*}x_n^\circ\end{align*} and yn\begin{align*}y_n^\circ\end{align*} are a linear pair Definition of a linear pair
3. xn\begin{align*}x_n^\circ\end{align*} and yn\begin{align*}y_n^\circ\end{align*} are supplementary Linear Pair Postulate
4. xn+yn=180\begin{align*}x_n^\circ+ y_n^\circ=180^\circ\end{align*} Definition of supplementary angles
5. (x1+x2++xn)+(y1+y2++yn)=180n\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 n\begin{align*}n-\end{align*}gon
6. (n2)180=(x1+x2++xn)\begin{align*}(n-2)180^\circ=(x_1^\circ+ x_2^\circ+\ldots+x_n^\circ)\end{align*} Polygon Sum Formula
7. 180n=(n2)180+(y1+y2++yn)\begin{align*}180^\circ n=(n-2)180^\circ+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} Substitution PoE
8. 180n=180n360+(y1+y2++yn)\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. 360=(y1+y2++yn)\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 y\begin{align*}y\end{align*}?

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

70+60+65+40+yy=360=125\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 360\begin{align*}360^\circ\end{align*}, so each angle is 360751.43\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 360\begin{align*}360^\circ\end{align*}.

#### Earlier Problem Revisited

The exterior angles of a regular polygon sum to 360\begin{align*}360^\circ\end{align*}. The measure of each exterior angle in a dodecagon (twelve-sided regular polygon) is 36012=30\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 360\begin{align*}360^\circ\end{align*} and by the given number of sides.

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

#### Example 2

100-gon

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

#### Example 3

36-gon

10\begin{align*}10^\circ\end{align*}

### Review

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

Find the measure of the missing variables:

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

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

1. octagon
2. nonagon
3. triangle
4. pentagon
5. 50-gon
6. heptagon
7. 34-gon
8. Challenge 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 360n\begin{align*}\frac{360^\circ}{n}\end{align*} and the other is 180(n2)180n\begin{align*}180^\circ - \frac{(n-2)180^\circ}{n}\end{align*} (180\begin{align*}180^\circ\end{align*} minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.
9. Angle Puzzle Find the measures of the lettered angles below given that m || n\begin{align*}m \ || \ n\end{align*}.

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Color Highlighted Text Notes

### Vocabulary Language: English

Exterior Angle Sum Theorem

Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to 360 degrees.

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