Exterior Angle Theorem

An angle is a geometrical figure with two rays and is generated from a common point. It is measured in degrees or radians and basically measures the degree of turn between the two sides of an angle. Now, the turns can be measured by finding out the interior angles or the exterior angle of straight lines. In this article, we shall be learning about exterior angles and the related concepts. 

Table of Content

• Introduction

• What is Exterior Angle

• Exterior Angle Property

• Exterior Angle Property of a Triangle

• Properties of Exterior Angle:

• Exterior Angle Theorem 

• Exterior Angle Theorem Proof

• Solved Examples

• Frequently asked questions

What is an Exterior Angle?

A triangle has three vertices or points. By joining these points we get three sides. The degree of turn between the sides when measured from the inside of the triangle or any object is its interior angle. Whereas, the angles or the degree of turn between the sides, when measured on the outer angles of an object, is its exterior angle. 

Exterior Angle Property

The exterior angle theorem is amongst the most basic theorems of triangles in geometry. Before we begin the discussion, let us have a look at what a triangle is. A polygon is called a plane figure that is bounded by the finite number of line segments for forming a closed figure.  The smallest polygon is known as a triangle since there are three line segments that are bound to it. The triangle is the smallest polygon which is bounded by three different line segments. It consists of three edges and three vertices. The exterior angle of the triangle is formed between any of the sides of the triangle and the extension of the adjacent side. We will learn in this lesson about the exterior angle theorem, exterior angle property, exterior angle theorem proof, and look at the examples.

Exterior Angle Property of a Triangle

Let us first learn about the exterior angle property before we learn about the exterior angle theorem.

An exterior angle of a triangle is equal to the angle formed between one side of the triangle and the extension of the adjacent side. Consider the figure given below.

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Properties of Exterior Angle

The properties of the exterior angle is given as follows:

1. The exterior angle of a given triangle equals the sum of the opposite interior angles of that triangle.

2. If an equivalent angle is taken at each vertex of the triangle, the exterior angles add to 360° in all the cases. In fact, this statement is true for any given convex polygon and not just triangles. 

Exterior Angle Theorem 

Let us learn more about the exterior angles and the exterior angle theorem in detail.

An exterior angle is an angle that is formed between one side of the polygon and the extension of the adjacent side.

In all the known polygons, there are two different sets of exterior angles, one that goes around the clockwise direction and the other that goes around the counterclockwise direction.

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You can notice here that the interior angle and its adjacent exterior angle both tend to form a linear pair and their sum adds up to 180°.

m∠1 + m∠2 = 180∘

The exterior angle theorem states that the sum total of all the remote interior angles of the triangle is equal to the non-adjacent exterior angle of that triangle. From the figure above, it means that m∠A + m∠B = m∠ACD. Given below is the proof of the exterior angle theorem. From the theorem’s proof, you would see that this theorem is the combination of both the Triangle Sum Theorem and the Linear Pair Postulate.

Exterior Angle Theorem Proof

Let us look at the exterior angle proof.

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Given is the △ABC with the exterior angle ∠ACD

We have to prove that m∠A + m∠B = m∠ACD

Given below is the proof:

Hence, it is proved that m∠A + m∠B = m∠ACD

Solved Examples

Take a look at the solved examples given below to understand the concept of the exterior angles and the exterior angle theorem.

Example 1

Find the measure of the unknown numbered interior and exterior angles in the given triangle below.

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Solution:

m∠1 + 92∘ = 180∘ through the Linear Pair Postulate

Hence, m∠1 = 88∘

m∠2 + 123∘ = 180∘ through the Linear Pair Postulate

Hence, m∠2 = 57∘

m∠1 + m∠2 + m∠3 =180∘through the Triangle Sum Theorem 

Hence, 88∘ + 57∘ + m∠3 = 180∘ and also m∠3 = 35∘

m∠3 + m∠4 = 180∘ through the Linear Pair Postulate

Hence, m∠4 = 145∘

Example 2

Determine the value of p in the triangle below

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Solution:

First, you need to find the missing exterior angle and you can call it x. Then set up an equation with the help of the Exterior Angle Sum Theorem

130∘ + 110∘ + x = 360∘

= x = 360∘ − 130∘ − 110∘

Hence, x = 120∘

x and p are the supplementary angles and add up to 180∘

x + p = 180∘

= 120∘ + p = 180∘

Hence, p = 60∘

Example 3

Determine m∠C

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Solution:

By using the exterior angle theorem, you get  m∠C + 16∘ = 121∘ 

By subtracting 16∘ from both the sides, you get m∠C = 105∘