The Angle Sum of a Triangle

Draw any triangle you like — tall, squat, lopsided — and add up its three inside angles. You always get the same total:

\angle A + \angle B + \angle C = 180^\circ

A flat half-turn. This one fact is the workhorse of triangle geometry, and the reason it is true comes straight from the alternate interior angles of parallel lines.

In every triangle, the three interior angles add up to a straight angle:

the sum is always \angle A + \angle B + \angle C = 180^\circ, whatever the triangle's shape or size;
so any two angles determine the third — it is 180^\circ minus the other two;
it follows from alternate interior angles: a line through the apex parallel to the base turns the three angles into angles on one straight line.

The proof, one step at a time

The trick is to draw a single extra line. Step through it: a line through the apex parallel to the base turns the triangle's angles into three angles on a straight line.

Because the alternate angles are equal, the angle on the line are the very same \angle A and \angle B from the triangle — and a straight line is 180^\circ. So \angle A + \angle C + \angle B = 180^\circ, which is exactly the triangle's three angles.

Using it

Know two angles of a triangle and you know the third — just subtract from 180^\circ. A triangle with angles 60^\circ and 50^\circ has a third angle of 180^\circ - 60^\circ - 50^\circ = 70^\circ.

Practise: chase the angles

Here is a triangle with the parallel-line construction already drawn. Fill in every angle you can — the base angles reappear at the top by alternate interior angles, and the three angles at the top sit on a straight line, which is the triangle's angle sum. Each problem needs both ideas. Refresh for a new one; Check explains every step.