The exterior angle as the sum of the non-adjacent interior angles

If one already knows that the sum of the interior angles of a triangle is $\pi$, the previous inequality is strengthened into a precise equality.

Theorem — Exterior angle (equality)

An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles: $\delta = \alpha + \beta.$

Proof

Let $\delta$ be the exterior angle adjacent to the angle $\gamma$. It is enough to put together two known facts.

1. I observe. $\delta + \gamma = \pi$ (because $\delta$ and $\gamma$ are supplementary).
2. I observe. $\alpha + \beta + \gamma = \pi$ (sum of the interior angles of a triangle).
3. I deduce. Comparing the two relations: $\delta = \pi - \gamma = \alpha + \beta$.

$\blacksquare$

Links

Topics: Euclidean geometry
Concepts: Exterior angle · Proof · Sum of the angles of a triangle
Skills: Proving