GeoProof — Triangle Angle Sum explorer

∠A + ∠B + ∠C = 180°

The Angle Sum of a Triangle

In every triangle — no matter its shape or size — the three interior angles add up to a straight angle:

∠A + ∠B + ∠C = 180°

Drag the vertices to make the triangle tall, flat, or lopsided; the total never budges from 180°.

The straight-line picture

Turn on "Fold onto a straight line" to lay the three angles next to each other at a single point. They fit together exactly along a straight line — which is what 180° means.

Why it's true

Draw a line through one vertex parallel to the opposite side. The two base angles reappear at that vertex as alternate angles with the parallel line, sitting on either side of the top angle — and together the three fill the straight line. (See the "Parallel Lines & Transversal" explorer for those equal alternate angles.)

Consequences

Each exterior angle equals the sum of the other two interior angles; an equiangular triangle has three 60° angles; and a right triangle's two non-right angles always add to 90°.

A note on flat space

This 180° total is a property of flat (Euclidean) geometry. On a sphere the angles of a triangle add to more than 180°, and on a saddle-shaped surface, less.