The Exterior Angle Theorem

Introduction

If one side of a triangle is extended, an exterior angle is formed.  An exterior angle of a triangle forms a linear pair with the adjacent interior angle. The two non-adjacent interior angles to the exterior angle are called its remote interior angles. In the figure below, \angle 1 and \angle 2 are remote angles of \angle 4.

exterior angle

In the second triangle above, we can see that the sum of the measures of the remote angles is equal to the measure of the exterior angle. That is, \angle 1 + \angle 2 = \angle 4. Is this observation always true? In this post, we prove that it is indeed true

Theorem

The measure of the exterior angle of a triangle is equal to the sum of the measure sf its interior angles.

Proof

m \angle 3 + m \angle 4 = 180 since the two angles are linear pair and therefore supplementary.

m \angle 4 = 180 - m \angle 3* by Addition Property of Equality

m \angle 1 + m \angle 2 + m \angle 3 = 180  because the angle sum of a triangle is 180 degrees.

m \angle 1 + m \angle 2 = 180 - m \angle 3 by Addition Property of Equality**

From * and **,  m \angle 1 + m \angle 2 and m \angle 4 both equal, 180 - m \angle 3,

m \angle 1 + m \angle 2 = m \angle 4. \blacksquare

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