**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, $latex \angle 1$ and $latex \angle 2$ are remote angles of $latex \angle 4$.

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, $latex \angle 1 + \angle 2 = \angle 4$. 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**

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

$latex m \angle 4 = 180 – m \angle 3$* by Addition Property of Equality

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

$latex m \angle 1 + m \angle 2 = 180 – m \angle 3$ by Addition Property of Equality**

From * and **, $latex m \angle 1 + m \angle 2$ and $latex m \angle 4$ both equal, $latex 180 – m \angle 3$,

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

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