# 6.10: Sum of the Exterior Angles of a Polygon

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Identify the exterior angles of convex polygons.
• Find the sums of exterior angles in convex polygons.

This lesson focuses on the exterior angles in a polygon. There is a surprising feature of the sum of the exterior angles in a polygon that will help you solve problems about regular polygons.

## Exterior Angles in Convex Polygons

Recall that interior means inside and that exterior means outside. So, an exterior angle is an angle on the outside of a polygon. An exterior angle is formed by extending a side of the polygon:

There are two possible exterior angles for any given vertex on a polygon. In the figure above, one set of exterior angles is shown, the set in the counter-clockwise direction. The other set of exterior angles would be formed by extending each side of the polygon in the opposite (clockwise) direction. However, it does not matter which exterior angles you use because their measurement will be the same on each vertex. Look closely at one vertex below, where we draw both of the possible exterior angles:

In the above diagrams, both exterior angles are drawn separately. On the next page, both exterior angles on a single vertex are drawn together:

As you can see, the two exterior angles at the same vertex are vertical angles.

Since vertical angles are congruent, the two exterior angles possible around a single vertex are congruent.

The clockwise exterior angle and the counter-clockwise exterior angle at the same vertex are _______________________________ .

Additionally, because the exterior angle will be a linear pair with its adjacent interior angle, it will always be supplementary to that interior angle.

As a reminder, supplementary angles have a sum of 180\begin{align*}180^\circ\end{align*}.

This means the exterior angle and interior angle at the same vertex will sum to ______\begin{align*}^\circ\end{align*}.

1. True or False: Vertical angles are supplementary.

2. True or False:

Exterior angles are on the outside of a polygon and are formed when you extend one side of the polygon.

3. Name the 2 sets of exterior angles:

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

4. Fill in the blank: Angles that form a linear pair add up to ________\begin{align*}^\circ\end{align*}.

5. Complete the sentence:

Since the interior angle and the exterior angle at the same vertex of a polygon form a linear pair, ...

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

Example 1

What is the measure of the exterior angle OKL\begin{align*}\angle OKL\end{align*} in the diagram below?

The interior angle JKL\begin{align*}\angle JKL\end{align*} is labeled as 45\begin{align*}45^\circ\end{align*}.

Notice that the interior angle and the exterior angle form a linear pair, meaning the two angles are supplementary – they add up to 180\begin{align*}180^\circ\end{align*}.

So, to find the measure of the exterior angle, subtract 45\begin{align*}45^\circ\end{align*} from 180\begin{align*}180^\circ\end{align*}:

18045=135\begin{align*}180^\circ - 45^\circ = 135^\circ\end{align*}

The measure of OKL\begin{align*}\angle OKL\end{align*} is 135\begin{align*}135^\circ\end{align*}.

## Summing Exterior Angles in Convex Polygons

By now you might expect that if you add up various angles in polygons, there will be some sort of pattern or rule.

For example, you already know that the sum of the interior angles in a triangle will always be 180\begin{align*}180^\circ\end{align*}.

From that fact, you learned that you can find the sums of the interior angles of any polygon with n\begin{align*}n\end{align*} sides using the expression 180(n2)\begin{align*}180^\circ(n - 2)\end{align*}.

There is also a rule for exterior angles in a polygon.

Let’s begin by looking at a triangle:

To find the exterior angles at each vertex, extend the segments and find angles supplementary to the interior angles:

The sum of these three exterior angles is:

150+120+90=360\begin{align*}150^\circ + 120^\circ + 90^\circ = 360^\circ\end{align*}

So, the exterior angles in this triangle will sum to 360\begin{align*}360^\circ\end{align*}.

Let’s see what happens with another shape.

To compare, examine the exterior angles of a rectangle:

In a rectangle, each interior angle measures ______ \begin{align*}^\circ\end{align*}.

Since exterior angles are supplementary to interior angles, all exterior angles in a rectangle will also measure ______ \begin{align*}^\circ\end{align*}.

Find the sum of the 4 exterior angles in a rectangle:

90+90+90+90=360\begin{align*}90^\circ + 90^\circ + 90^\circ + 90^\circ = 360^\circ\end{align*}

So, the sum of the exterior angles in a rectangle is also 360\begin{align*}360^\circ\end{align*}.

In fact, the sum of the exterior angles in any convex polygon will always be 360\begin{align*}360^\circ\end{align*}.

It does not matter how many sides the polygon has, the sum will always be 360\begin{align*}360^\circ\end{align*}.

Exterior Angle Sum

The sum of the exterior angles of any convex polygon is 360\begin{align*}360^\circ\end{align*}.

No matter how many sides a polygon has, the sum of its __________________________ angles is equal to _______ \begin{align*}^\circ\end{align*}.

We can prove this using algebra and the facts that at any vertex the sum of the interior and one of the exterior angles is always 180\begin{align*}180^\circ\end{align*}, and the sum of all interior angles in a polygon is 180(n2)\begin{align*}180^\circ(n - 2)\end{align*}. The proof is on the next page.

Proof:

At any vertex of a polygon the exterior angle and the interior angle sum to 180\begin{align*}180^\circ\end{align*}. So, adding up all of the exterior angles and interior angles gives a total of 180\begin{align*}180^\circ\end{align*} times the number of vertices:

(Sum of Exterior Angles)+(Sum of Interior Angles)=180n\begin{align*}(\text{Sum of Exterior Angles}) + (\text{Sum of Interior Angles}) = 180^\circ n\end{align*}

On the other hand, we already saw that the sum of the interior angles was:

(Sum of Interior Angles)=180(n2)=180n360 (using the Distributive Property)\begin{align*}(\text{Sum of Interior Angles}) & = 180^\circ (n - 2)\\ & = 180^\circ n - 360^\circ \ (\text{using the Distributive Property})\end{align*}

Putting these together we have:

180n=(Sum of Exterior Angles)+(Sum of Interior Angles)=(180n360)+(Sum of Exterior Angles)\begin{align*}180^\circ n & = (\text{Sum of Exterior Angles}) + (\text{Sum of Interior Angles})\\ & = (180^\circ n - 360^\circ) + (\text{Sum of Exterior Angles})\end{align*}

Subtract (180n360)\begin{align*}(180^\circ n - 360^\circ)\end{align*} from both sides and:

360=(Sum of Exterior Angles)\begin{align*}360^\circ = (\text{Sum of Exterior Angles})\end{align*}

1. True or False:

In a polygon, an interior angle and one of the exterior angles at the same vertex are supplementary.

2. True or False: Supplementary angles add up to 90\begin{align*}90^\circ\end{align*}.

3. Fill in the blanks:

The sum of all interior __________________ in a polygon with n\begin{align*}n\end{align*} sides is equal to _______(n2)\begin{align*}^\circ \cdot (n - 2)\end{align*}.

4. What is the rule for the sum of exterior angles in a polygon? Describe.

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

Example 2

What is mQXZ\begin{align*}m \angle QXZ\end{align*} in the diagram below?

QXZ\begin{align*}\angle QXZ\end{align*} in the diagram is an exterior angle. So, we need to find the measure of one exterior angle on a polygon given the measures of all of the others.

4 of the 5 exterior angles on this polygon have their measurements labeled:

,,,\begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;} ^\circ, \underline{\;\;\;\;\;\;\;\;\;\;\;\;} ^\circ, \underline{\;\;\;\;\;\;\;\;\;\;\;\;} ^\circ,\end{align*} and \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;} ^\circ\end{align*}

We know that the sum of the exterior angles on a polygon must be equal to 360\begin{align*}360^\circ\end{align*}, regardless of how many sides the shape has. So, we can set up an equation where we set all of the exterior angles shown (including mQXZ\begin{align*}m \angle QXZ\end{align*}) summed and equal to 360\begin{align*}360^\circ\end{align*}:

70+60+65+40+mQXZ235+mQXZ235mQXZ = 360 = 360235 = 125\begin{align*}70^\circ + 60^\circ + 65^\circ + 40^\circ + m \angle QXZ & \ = \ 360^\circ\\ 235^\circ + m \angle QXZ & \ = \ 360^\circ\\ -235^\circ{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} & {\;\;\;\;}- 235^\circ\\ m \angle QXZ & \ = \ 125^\circ\end{align*}

The measure of the missing exterior angle (QXZ\begin{align*}\angle QXZ\end{align*}) is 125\begin{align*}125^\circ\end{align*}.

We can check that our answer is reasonable by inspecting the diagram and checking whether the angle in question is acute, right, or obtuse. Since the angle should be obtuse, 125\begin{align*}125^\circ\end{align*} is a reasonable answer (assuming the diagram is accurate).

The sum of the exterior angles of any convex polygon is equal to ________\begin{align*}^\circ\end{align*}.

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