The Exterior Angle of a Triangle

Take a triangle and extend one side past a corner. The angle you open up between the extension and the next side is an exterior angle of the triangle. It has a beautifully simple value — it equals the two angles at the other two corners, added together:

\text{exterior angle} = \text{the two remote interior angles}

("Remote" just means the two corners the exterior angle is not touching.) This follows directly from the angle sum of a triangle.

Extend one side of a triangle to make an exterior angle at a vertex. Then:

the exterior angle equals the sum of the two remote interior angles — \angle\text{ext} = \angle A + \angle C for the exterior angle at B;
it also makes a straight line with the interior angle next to it, so \angle\text{ext} + \angle B = 180^\circ;
so an exterior angle is always bigger than either remote interior angle.

Why it works

Two facts you already know do all the work: a straight line is 180^\circ, and a triangle's angles add to 180^\circ. Step through it.

In symbols: the interior angle at B is 180^\circ - (\angle A + \angle C) by the angle sum, and the exterior angle is 180^\circ - \angle B by the straight line — so the exterior angle is \angle A + \angle C.

Practise: chase the angles

A triangle with one side extended. Fill in every angle you can — use the exterior-angle rule, angles on a straight line, and the triangle's angle sum — ending with the highlighted one. Refresh for a new triangle; Check explains each step.