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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

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

  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 \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 

  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 \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:

  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 \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.
  9. Angle Puzzle Find the measures of the lettered angles below given that \begin{align*}m \ || \ n\end{align*}.

Review (Answers)

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

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Vocabulary

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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