Geometry (A): READING: Exterior Angles Reading

Exterior Angles

Learning Objectives

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

Introduction

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.

As you can tell, there are two possible exterior angles for any given vertex on a polygon. In the figure above we only showed one set of exterior angles; the other set would be formed by extending each side in the opposite (clockwise) direction. However, it doesn’t matter which exterior angles you use because on each vertex their measurement will be the same. Let’s look closely at one vertex, and draw both of the exterior angles that are possible.

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.

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 $180^\circ$ .

Example 1

What is the measure of the exterior angle $\angle {OKL}$ in the diagram below?

The interior angle is labeled as $45^\circ$ . Since you need to find the exterior angle, notice that the interior angle and the exterior angle form a linear pair. Therefore the two angles are supplementary—they sum to $180^\circ$ . So, to find the measure of the exterior angle, subtract $45^\circ$ from $180^\circ$ .

$180 - 45 = 135$

The measure of $\angle {OKL}$ is $135^\circ$ .

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 know that the sum of the interior angles of a triangle will always be $180^\circ$ . From that fact, you have learned that you can find the sums of the interior angles of any polygons with $n$ sides using the expression $180(n-2)$ . 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:

$150^\circ + 120^\circ + 90^\circ = 360^\circ$

So, the exterior angles in this triangle will sum to $360^\circ$ .

To compare, examine the exterior angles of a rectangle.

In a rectangle, each interior angle measures $90^\circ$ . Since exterior angles are supplementary to interior angles, all exterior angles in a rectangle will also measure $90^\circ$ .

Find the sum of the four exterior angles in a rectangle.

$90^\circ + 90^\circ + 90^\circ + 90^\circ = 360^\circ$

So, the sum of the exterior angles in a rectangle is also $360^\circ$ .

In fact, the sum of the exterior angles in any convex polygon will always be $360^\circ$ . It doesn’t matter how many sides the polygon has, the sum will always be $360^\circ$ .

We can prove this using algebra as well as the facts that at any vertex the sum of the interior and one of the exterior angles is always $180^\circ$ , and the sum of all interior angles in a polygon is $180(n-2)$ .

Exterior Angle Sum: The sum of the exterior angles of any convex polygon is $360^\circ$

Proof. At any vertex of a polygon the exterior angle and the interior angle sum to $180^\circ$ . So summing all of the exterior angles and interior angles gives a total of 180 degrees times the number of vertices:

$\text{(Sum of Exterior Angles)} + \text{(Sum of Interior Angles)} = 180^\circ n .$

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

$\text{(Sum of Interior Angles)} = 180(n-2) = 180^\circ n - 360^\circ.$

Putting these together we have

$180 n &= \text{(Sum of Exterior Angles)} + \text{(Sum of Interior Angles)} \\ &= (180 n - 360) + \text{(Sum of Exterior Angles)}\\ 360 &= \text{(Sum of Exterior Angles)}$

Example 2

What is $m \angle {ZXQ}$ in the diagram below?

$\angle {ZXQ}$ in the diagram is marked as 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. We know that the sum of the exterior angles on a polygon must be equal to $360^\circ$ , 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 $m \angle {ZXQ}$ ) summed and equal to $180^\circ$ . Using subtraction, we can find the value of $X$ .

$70^\circ + 60^\circ + 65^\circ + 40^\circ + m \angle {ZXQ} &= 360^\circ \\ 235^\circ + m \angle {ZXQ} &= 360^\circ \\ m \angle {ZXQ} &= 360^\circ - 235^\circ \\ m \angle {ZXQ} &= 125^\circ$

The measure of the missing exterior angle is $125^\circ$ .

We can verify 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, $125^\circ$ is a reasonable answer (assuming the diagram is accurate).

Lesson Summary

In this lesson, we explored exterior angles in polygons. Specifically, we have learned:

How to identify the exterior angles of convex polygons.
How to find the sums of exterior angles in convex polygons.

We have also shown one example of how knowing the sum of the exterior angles can help you find the measure of particular exterior angles.

The following questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

For exercises 1-3, find the measure of each of the labeled angles in the diagram.

$x =\underline{\;\;\;\;\;\;\;}, y =\underline{\;\;\;\;\;\;\;}$
$w =\underline{\;\;\;\;\;\;\;}, x =\underline{\;\;\;\;\;\;\;}, y =\underline{\;\;\;\;\;\;\;}, z =\underline{\;\;\;\;\;\;\;}$
$a =\underline{\;\;\;\;\;\;\;}, b =\underline{\;\;\;\;\;\;\;}$
Draw an equilateral triangle with one set of exterior angles highlighted. What is the measure of each exterior angle? What is the sum of the measures of the three exterior angles in an equilateral triangle?
Recall that a regular polygon is a polygon with congruent sides and congruent angles. What is the measure of each interior angle in a regular octagon?
How can you use your answer to 5 to find the measure of each exterior angle in a regular octagon? Draw a sketch to justify your answer.
Use your answer to 6 to find the sum of the measures of the exterior angles of an octagon.

Complete the following table assuming each polygon is a regular polygon. Note: This is similar to a previous exercise with more columns—you can use your answer to that question to help you with this one.

Regular Polygon name	Number of sides	Sum of measures of interior angles	Measure of each interior angle	Measure of each exterior angle	Sum of measures of exterior angles
triangle
	$4$
	$5$
	$6$
	$7$
octagon
decagon
		$1,800^\circ$
	$n$

Each exterior angle forms a linear pair with its adjacent internal angle. In a regular polygon, you can use two different formulas to find the measure of each exterior angle. One way is to compute $180^\circ -$ (measure of each interior angle) $\ldots$ in symbols $180 - \frac{180(n - 2)} {n}.$

Alternatively, you can use the fact that all $n$ exterior angles in an $n-$ gon sum to $360^\circ$ and find the measure of each exterior angle with by dividing the sum by $n$ . Again, in symbols this is $\frac{360} {n}$

Use algebra to show these two expressions are equivalent.

Review Answers

$x = 52^\circ$ , $y = 128^\circ$
$w = 70^\circ, x = 70^\circ, y = 110^\circ, z = 90^\circ$
$a = 107.5^\circ , b = 72.5^\circ$
Below is a sample sketch.

Each exterior angle measures $120^\circ$ , the sum of the three exterior angles is $360^\circ$

Sum of the angles is $180(8 - 2) = 1080^\circ$ . So, each angle measures $\frac{1080} {8} = 135^\circ$
Since each exterior angle forms a linear pair with its adjacent interior angle, we can find the measure of each exterior angle with $180 - 135 = 45^\circ$
$45(8) = 360^\circ$

Regular Polygon name	Number of sides	Sum of measures of interior angles	Measure of each interior angle	Measure of each exterior angle	Sum of measures of exterior angles
triangle	$3$	$180^\circ$	$60^\circ$	$120^\circ$	$360^\circ$
square	$4$	$360^\circ$	$90^\circ$	$90^\circ$	$360^\circ$
pentagon	$5$	$540^\circ$	$72^\circ$	$108^\circ$	$360^\circ$
hexagon	$6$	$720^\circ$	$60^\circ$	$120^\circ$	$360^\circ$
heptagon	$7$	$900^\circ$	$128.57^\circ$	$51.43^\circ$	$360^\circ$
octagon	$8$	$1,080^\circ$	$135^\circ$	$45^\circ$	$360^\circ$
decagon	$10$	$1,440^\circ$	$144^\circ$	$36^\circ$	$360^\circ$
dodecagon	$12$	$1,800^\circ$	$150^\circ$	$30^\circ$	$360^\circ$
$n-$ gon	$n$	$180 (n - 2)^\circ$	${\frac{180 (n - 2)} {n}}^\circ$	${\frac{360} {n}}^\circ$	$360^\circ$

One possible answer.
$180 - \frac{180(n - 2)} {n} &= \frac{180 n} {n} - \frac{180 (n - 2)} {n}\\ &= \frac{180 n - 180 (n - 2)} {n}\\ &= \frac{180 n - 180 n + 360} {n}\\ &= \frac{360} {n}$

Last modified: Monday, July 12, 2010, 5:47 PM