Introduction

Consider the polygon ABCD , where the side \overline{AB} is extended beyond the vertex B.

We call \color{blue}\angle{CBE} the exterior angle of the vertex B .

Given a polygon's vertex, the interior and exterior angles are supplementary. So, for the vertex B, we have: \underbrace{\color{YellowOrange}m\angle{ABC}}_{\large\text{Interior}} \:+\: \underbrace{\color{blue}m\angle{CBE}}_{\large\text{Exterior}} = 180^\circ

We have the following theorem:

The measures of the exterior angles of any simple polygon sum to 360^\circ.

We'll show why this theorem is true at the end of the lesson.

Let's label the exterior angles of our polygon below:

According to our theorem, these four angles sum to 360^\circ. a+b+c+d = 360^\circ

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Example: Identifying Exterior Angles

Consider the hexagon shown below. Draw an exterior angle of \angle B.

EXPLANATION

To construct an exterior angle of \angle B, we extend \overline{AB} past B. The angle between \overline{BC} and the extension of \overline{AB} is an exterior angle of \angle B.

Another way to construct an exterior angle of \angle B is to extend \overline{CB} past B. The angle between \overline{AB} and the extension of \overline{CB} is also an exterior angle of \angle B.

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Practice: Identifying Exterior Angles

Consider the quadrilateral shown above. Which of the following shows an exterior angle of $\angle C?$

a
b
c
d
e
Practice: Identifying Exterior Angles

Consider the pentagon shown above. Which of the following shows an exterior angle of $\angle E?$

a
b
c
d
e
Example: Calculating Exterior Angles

In the figure above, determine the measure of the exterior angle of \angle{F}.

EXPLANATION

Let us call x the exterior angle of \angle F. Since the sum of the exterior angles of a polygon is 360^{\circ} , we have

\begin{align*} 100^\circ+60^\circ+78^\circ+62^\circ+x &=360^\circ\\[5pt] 300^\circ+x &=360^\circ\\[5pt] x &= 360^\circ - 300^\circ \\[5pt] x &= 60^\circ. \end{align*}

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Practice: Calculating Exterior Angles

For the quadrilateral above, the measure of the exterior angle of $\angle D$ is

a
b
c
d
e
Practice: Calculating Exterior Angles

For the pentagon above, determine the measure of the exterior angle of $\angle B.$

a
 $90^\circ$
b
 $100^\circ$
c
 $60^\circ$
d
 $70^\circ$
e
 $80^\circ$
Example: Determining an Unknown Quantity Using Exterior Angles

Find the value of x.

EXPLANATION

Note that the measure of the exterior angle at A is 180^\circ - 90^\circ=90^\circ and the measure of the exterior angle at D is 180^\circ - 2x.

Since the sum of the exterior angles of a polygon is 360^{\circ}, we have \begin{align*} 90^\circ + (x+10^\circ) + (180^\circ - 2x) + x + (x+10^\circ) & = 360^\circ\\[5pt] x + 290^\circ &= 360^\circ\\[5pt] x &= 70^\circ. \end{align*}

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Practice: Determining an Unknown Quantity Using Exterior Angles

Find the value of $x.$

a
 $45^\circ$
b
 $84^\circ$
c
 $34^\circ$
d
 $46^\circ$
e
 $25^\circ$
Practice: Determining an Unknown Quantity Using Exterior Angles

Given the pentagon above, we have $x =$

a
 $50^\circ$
b
 $35^\circ$
c
 $55^\circ$
d
 $40^\circ$
e
 $45^\circ$
Example: Calculating an Exterior Angle by Determining an Unknown Quantity

For the quadrilateral above, determine the measure of the exterior angle of \angle A.

EXPLANATION

Since the sum of the exterior angles of a polygon is 360^{\circ} , we have \begin{align*} (12x-5^\circ) + (6x+5^\circ) + (10x) + (9x-10^\circ) &=360^\circ\\[5pt] 37x - 10^\circ & = 360^\circ\\[5pt] 37x & = 360^\circ + 10^\circ\\[5pt] 37x & = 370^\circ\\[5pt] x & = 10^\circ. \end{align*}

Therefore, the measure of the exterior angle of \angle A equals \begin{align*} 12x-5^\circ & = 12(10^\circ)-5^\circ \\[5pt] & = 120^\circ - 5^\circ\\[5pt] & = 115^\circ. \end{align*}

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Practice: Calculating an Exterior Angle by Determining an Unknown Quantity

For the quadrilateral above, the measure of the exterior angle of $\angle B$ is

a
b
c
d
e
Practice: Calculating an Exterior Angle by Determining an Unknown Quantity

For the pentagon above, determine the measure of the exterior angle of $\angle A.$

a
 $43^\circ$
b
 $53^\circ$
c
 $55^\circ$
d
 $65^\circ$
e
 $45^\circ$
Proving the Theorem

Why do the exterior angles of a polygon always sum to 360^\circ?

Suppose we have an n -sided polygon. Then, we have the following:

  • There are n pairs of interior-exterior angles, each pair being supplementary (i.e., their measures add to 180^\circ).

  • Therefore, the sum of the measures of all interior-exterior pairs equals {\color{blue}n\cdot 180^\circ}.

  • In a previous lesson, we saw that the sum of the interior angles equals {\color{red}{(n-2)\cdot 180^\circ}}.

Therefore, to get the sum of the exterior angles, we subtract {\color{red}{(n-2)\cdot 180^\circ}} from {\color{blue}n\cdot 180^\circ} as follows: \begin{align*} \require{cancel} \textrm{Sum of Exterior Angles} & = {\color{blue}n\cdot 180^\circ} - {\color{red}{(n-2)\cdot 180^\circ}}\\[5pt] &= n\cdot 180^\circ -n\cdot 180^\circ + 2\cdot 180^\circ \\[5pt] &= \cancel{n\cdot 180^\circ} -\cancel{n\cdot 180^\circ} + 2\cdot 180^\circ \\[5pt] &=2\cdot 180^\circ \\[5pt] &= 360^\circ \end{align*}

This proves that the sum of the exterior angles of a (simple) polygon equals 360^\circ.

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