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Exterior Angles in Convex Polygons

Angles on the outside of a polygon formed by extending a side.

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Exterior Angles in Convex Polygons

What if you were given a twelve-sided regular polygon? How could you determine the measure of each of its exterior angles? After completing this Concept, you'll be able to use the Exterior Angle Sum Theorem to solve problems like this one.

Watch This

Watch the second half of this video.

Guidance

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.

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

Example A

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

Example B

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*}.

Example C

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*}.

Watch this video for help with the Examples above.

Concept 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*}.

Vocabulary

An exterior angle is an angle that is formed by extending a side of the polygon. A regular polygon is a polygon in which all of its sides and all of its angles are congruent.

Guided Practice

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

1. 12-gon

2. 100-gon

3. 36-gon

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

1. 30\begin{align*}30^\circ\end{align*}

2. 3.6\begin{align*}3.6^\circ\end{align*}

3. 10\begin{align*}10^\circ\end{align*}

Practice

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*}.

Vocabulary Language: English

Exterior Angle Sum Theorem

Exterior Angle Sum Theorem

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