# Exterior Angle Theorem

Exterior Angle Theorem
Go back to  'Triangles'

Some common facts about the triangle that are already known to us-

• A triangle has 3 internal angles which always add to 180 degrees.
• It is a polygon with the least number of sides (three sides).
• It has 6 external angles.

Let us explore the exterior angle theorem as we scroll down. Also, there is a simulator for you to explore the exterior angle theorem, and try your hand at solving the interactive questions at the end.

## Lesson Plan

 1 What Is the Exterior Angle Theorem? 2 Important Notes on Exterior Angle Theorem 3 Challenging Question on Exterior Angle Theorem 4 Solved Examples on Exterior Angle Theorem 5 Interactive Questions on Exterior Angle Theorem

## What Is the Exterior Angle Theorem?

An exterior angle of a triangle is formed when any side of a triangle is extended. $$\angle D$$ is an exterior angle for the given triangle.

Exterior Angle Theorem :

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle. The remote interior angles are also termed as opposite interior angles.

Why Does This Theorem Work?

Here's why,

## What Is Exterior Angle Inequality Theorem?

The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles. This condition is satisfied by all the six external angles of a triangle.

## How to Use Exterior Angle Theorem?

To use the exterior angle theorem in a triangle we first need to identify the exterior angle and then the associated two remote interior angles of the triangle.

A common mistake of considering the adjacent interior angle should be avoided.

The theorem can then be used to find the measure of an unknown angle.

Note that an exterior angle is supplementary to its adjacent interior angle as they form a linear pair of angles.

Example:

Also, $$\angle C = 180^{\circ} - 110^{\circ} = 70^{\circ}$$

Important Notes
• The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle.
• The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles.
• The exterior angle and the adjacent interior angle are supplementary.

## Where Should We Use Exterior Angle Theorem?

Exterior angle theorem could be used to determine the measures of the unknown interior and exterior angles of a triangle.

Let us see a couple of examples to understand the use of the exterior angle theorem.

Example 1: Find the value of $$\angle x$$.

Since, $$\angle x$$ and given $$92^{\circ}$$ are supplementary,

$$\angle x + 92^{\circ} = 180^{\circ}$$

$$\angle x = 180^{\circ} - 92^{\circ} = 88^{\circ}$$

Applying the exterior angle theorem,

$$\angle y+ 41^{\circ} = 88^{\circ}$$

$$\angle y= 88^{\circ} - 41^{\circ} = 47^{\circ}$$

Challenging Question
1. Can you find the measure of $$\angle x$$ and $$\angle y$$?

## Solved Examples

 Example 1

Natalie has just learned the exterior angle theorem but she is finding it difficult to find the value of $$x$$. Can you help her with the same?

Solution

\begin{align*} \angle x + 3x &= 160^{\circ}\\ \angle 4x &= 160^{\circ}\\ \angle x &= \dfrac {160}{4}\\ \angle x&= 40^{\circ} \end{align*}

 $$\therefore$$ $$\angle x = 40^{\circ}$$
 Example 2

Martha is struggling to find the measures of all the interior angles of the given triangle. Can you help her?

Solution

As per the exterior angle theorem,

\begin{align*} (a-16^{\circ}) + (a+22^{\circ}) &= (a+62^{\circ})\\ 2a - a + 6^{\circ} &= 62^{\circ}\\ a &= 56^{\circ} \end{align*}

The exterior angle of the triangle will be $$(a + 62^{\circ}) = 56^{\circ} + 62^{\circ} = 118^{\circ}$$

The three internal angles can be found out by substituting the value of $$a = 54$$,

1. $$a-16^{\circ} = 56^{\circ} - 16^{\circ} = 40^{\circ}$$

2. $$a+22^{\circ} = 56^{\circ} + 22^{\circ} = 78^{\circ}$$

3. The third internal angle of the triangle can be given as

$$180^{\circ} - \text{ exterior angle} = 180^{\circ} - 118^{\circ} = 62^{\circ}$$ ........... [ $$\because$$ exterior angle and the adjacent interior angle are supplementary]

 $$\therefore$$ The measures of the angles of the given triangle are $$40^{\circ}$$, $$78^{\circ}$$, and $$62^{\circ}$$.
 Example 3

Ryan is studying the applications of the exterior angle theorem. Consider the following figure. Can you help him find the value of $$\angle{BAC}$$?

Solution

Solving the linear pair at vertex $$\text D$$,

\begin{align*} \angle{ADC} \ + \ \angle{ADE} &= 180^\circ \\ \angle 1 &= 180^\circ - 150^\circ \\ &= 30^\circ \end{align*}

Also, using angle sum property of a triangle, for $$\triangle{ACD}$$,

\begin{align*} \angle 1 + \angle 1 + \angle{ACD} &= 180^\circ \\ \angle{ACD} &= 180^\circ - 2 \times \angle 1 \\ \angle{ACD} &= 180^\circ - 60^\circ &= 120^\circ \end{align*}

$$\angle{ACD}$$ is the exterior angle for the$$\triangle{ABC}$$.

Thus, using exterior angle theorem for $$\triangle{ABC}$$,

\begin{align*} \angle{ACD} &= \angle{ABC} + \angle{BAC} \\ \angle{BAC} &= \angle{ACD} - \angle{ABC} \\ \angle{BAC} &= 120^\circ - 60^\circ \\ &= 60^\circ \end{align*}

 $$\therefore \angle{BAC} = 60^\circ$$

## Interactive Questions on Exterior Angle Theorem

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of exterior angle theorem. The math journey around exterior angle theorem starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What is the exterior angle theorem?

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle. The remote interior angles are also called opposite interior angles.

## 2. How to do the exterior angle theorem?

To use the exterior angle theorem in a triangle we first need to identify the exterior angle and then the associated two remote interior angles of the triangle.

A common mistake of considering the adjacent interior angle should be avoided.

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle. The remote interior angles are also termed opposite interior angles.

## 3. What are exterior angles?

An exterior angle of a triangle is formed when any side of a triangle is extended. $$\angle D$$ is an exterior angle for the given triangle.

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