Introduction

Consider the following triangle, where the side \overline{AC} has been extended beyond the vertex C.



The resulting angle \angle{BCD} is called the exterior angle at vertex C.

Exterior angles have the following properties:

  • An exterior angle and its corresponding interior angle form a linear pair and are therefore supplementary. Here, we have \underbrace{\color{YellowOrange}m\angle{BCD}}_{\large\text{Exterior}} \:+\: \underbrace{\color{blue}m\angle{ACB}}_{\large\text{Interior}} = 180^\circ

  • The measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles. This is called the exterior angles theorem. To see why it is true, notice that since the internal angles of a triangle sum to 180^\circ, we have \begin{align*} m\angle{A} +m\angle{B} +m\angle{C} &= 180^\circ \\ m\angle{A} + m\angle{C} &= \underbrace{180^\circ-m\angle{B}}_{\large\textrm{Exterior angle at [math]B[/math]}}. \end{align*}

  • At each vertex of a triangle, there are two exterior angles. They are vertical angles, and consequently they have the same measure.


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Example: Finding the Value of a Variable Given an Exterior Angle and Two Internal Angles

Find the value of x in the following figure.



EXPLANATION

Method 1

Let's start by finding the value of the missing interior angle at B. We use the fact that the interior and exterior angles are supplementary:

\begin{align*} 73^\circ + m\angle{B} &= 180^\circ \\[3pt] m\angle{B} &= 180^\circ-73^\circ \\[3pt] m\angle{B} &= 107^\circ \end{align*}

Now, we calculate x using the fact that the interior angles of a triangle sum to 180^\circ\mathbin{:} \begin{align*} m\angle{A} + m\angle{B} + m\angle{C} &= 180^\circ \\[3pt] 41^\circ + 107^\circ +2x &= 180^\circ \\[3pt] 148^\circ+2x &= 180^\circ \\[3pt] 2x &= {32}^\circ \\[3pt] x &= 16^\circ \end{align*}

Method 2

We could rewrite the equation m\angle{A} +m\angle{B} +m\angle{C} = 180^\circ as

\begin{align*} \underbrace{180^\circ-m\angle{B}}_{\large\textrm{Exterior angle at [math]B[/math]}} \!\!\!= m\angle{A} + m\angle{C}. \end{align*}

Substituting, we can solve for x\mathbin{:}

\begin{align*} \underbrace{180^\circ-m\angle{B}}_{\large\textrm{Exterior angle at [math]B[/math]}} \!\!\!&= m\angle{A} + m\angle{C} \\[3pt] 73^\circ &= 41^\circ + 2x \\[3pt] 32^\circ &= 2x \\[3pt] 16^\circ &= x \end{align*}

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Practice: Finding the Value of a Variable Given an Exterior Angle and Two Internal Angles

Find the value of $x$ in the figure above.

a
 $37^\circ$
b
 $33^\circ$
c
 $32^\circ$
d
 $38^\circ$
e
 $35^\circ$
Practice: Finding the Value of a Variable Given an Exterior Angle and Two Internal Angles

Calculate the value of $x$ in the figure above.

a
 $18^\circ$
b
 $21^\circ$
c
 $19^\circ$
d
 $20^\circ$
e
 $22^\circ$
The Sum of Exterior Angles

Yet another important fact about exterior angles is:

The sum of the measures of the exterior angles of a triangle is always equal to 360^\circ.

So, for the triangle below, we have

x+y+z = 360^\circ.

To see why, first recall that the measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles. So, we have \begin{align*} x &= m\angle B + m\angle C \\ y &= m\angle A + m\angle C \\ z &= m\angle A + m\angle B . \end{align*}

Now, remember that the internal angles of a triangle sum to 180^\circ. Then we have \begin{align*} x + y + z &= (m\angle B + m\angle C) + (m\angle A + m\angle C) + (m\angle A + m\angle B) \\[5pt] &= 2 m\angle A + 2 m\angle B + 2m\angle C \\[5pt] &= 2 \left( m\angle A + m\angle B + m\angle C \right) \\[5pt] &= 2 \left( 180^\circ \right) \\[5pt] &= 360^\circ. \end{align*}

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Example: Finding the Value of a Variable Given the Exterior Angles

Find the value of x in the figure below.


EXPLANATION

Since the exterior angles of a triangle sum to 360^{\circ} , we have \begin{align*} 145^{\circ} + x+(2x-25^\circ)&=360^\circ \\[3pt] 120^{\circ} + 3x &= 360^\circ \\[3pt] 3x & = 240^\circ \\[3pt] x& = 80^\circ. \end{align*}

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Practice: Finding the Value of a Variable Given the Exterior Angles

Find the value of $x$ in the figure above.

a
 $130^\circ$
b
 $145^\circ$
c
 $150^\circ$
d
 $135^\circ$
e
 $140^\circ$
Practice: Finding the Value of a Variable Given the Exterior Angles

Find the value of $x$ in the figure above.

a
 $67^\circ$
b
 $75^\circ$
c
 $65^\circ$
d
 $80^\circ$
e
 $70^\circ$
Example: Calculating the Measure of an Interior Angle Given the Exterior Angles

Given a triangle \triangle ABC , let the exterior angles at {A} and {B} be equal 120^\circ and 100^\circ, respectively. What is the measure of the angle \angle C?

EXPLANATION

Let's start by drawing the corresponding triangle.



Let z be the measure of the exterior angle at {C} pictured above. Since the sum of the exterior angles of a triangle is 360^{\circ} , we have \begin{align*} 120^{\circ} + 100^\circ + z &= 360^\circ \\[5pt] 220^{\circ} + z &= 360^\circ \\[5pt] z &= 140^\circ. \end{align*}

Now, we can find the value of the missing interior angle at C\mathbin{:} \begin{align*} m\angle{C} &= 180^\circ - z \\[5pt] &= 180^\circ - 140^\circ \\[5pt] &= 40^\circ \end{align*}

Therefore, m\angle C = 40^\circ.

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Practice: Calculating the Measure of an Interior Angle Given the Exterior Angles

Consider a triangle $\triangle ABC$, where an exterior angle at ${A}$ is $128^{\circ}$ and an exterior angle at ${B}$ is $115^{\circ}.$ What is the measure of the interior angle $C?$

a
 $60^{\circ}$
b
 $67^{\circ}$
c
 $65^{\circ}$
d
 $63^{\circ}$
e
 $61^{\circ}$
Practice: Calculating the Measure of an Interior Angle Given the Exterior Angles

Consider a triangle $\triangle ABC$, where an exterior angle at ${A}$ is $115^{\circ}$ and an exterior angle at ${B}$ is $97^{\circ}.$ What is the measure of the interior angle $C?$

a
 $32^{\circ}$
b
 $28^{\circ}$
c
 $30^{\circ}$
d
 $37^{\circ}$
e
 $35^{\circ}$
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