Exterior Angle Theorem

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 look at what is a triangle. 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 is 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 what is 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 add 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:

Statement 

Reason

△ABC with the exterior angle ∠ACD

It is given 

m∠A + m∠B + m∠ACB = 180

According to the known Triangle Sum Theorem

m∠ACB + m∠ACD = 180

According to the known Linear Pair Postulate

m∠A + m∠B + m∠ACB = m∠ACB + m∠ACD

According to the known Transitive PoE

m∠A + m∠B = m∠ACD

According to the known Subtraction PoE


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 = 180through the Linear Pair Postulate

Hence, m∠2 = 57

m∠1 + m∠2 + m∠3 =180through 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

FAQ (Frequently Asked Questions)

1. What is The Statement Of The Exterior Angle Theorem?

The exterior angle of a triangle is formed between any side of that given triangle and the extension of the triangle’s adjacent side.

The exterior angle theorem states that the exterior angle of the given triangle equals to the sum of the two opposite interior angles present in that triangle.

The following diagram shows the exterior angle theorem.

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2. What is The Use Of The Exterior Angle Theorem?

The exterior angle of a given triangle is formed when a side is extended outwards. The exterior angle theorem states that the exterior angle of the given triangle is equal to the sum total of its opposite interior angles.

You can derive the exterior angle theorem with the help of the information that

  1. The angles on the straight line add up to 180°

  2. The interior angles of the given triangle add up to 180°

3. What is The Sum Of All The Exterior Angles Of A Triangle?

The exterior angle of a given polygon is the angle outside the polygon which is formed by one of its sides and the extension of its adjacent side. The sum of all the exterior angles of the given polygon results to 360° in all the cases irrespective of the number of sides of the polygon. Hence, the answer is 360°.