# 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 $\angle 4$.

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, $180 - m \angle 3$,

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

## 3 thoughts on “The Exterior Angle Theorem”

1. This should be proven in neutral geometry. You cannot assume triangles have angle sum 180 in neutral geometry. That’s Euclidean geometry only.