GeoProof — Five-Pointed Star Angle Sum

∠1 + ∠2 + ∠3 + ∠4 + ∠5 = 180°

The Star's Angle Sum

Take any five-pointed star drawn in one stroke (a pentagram). Add up the angles at its five points:

∠1 + ∠2 + ∠3 + ∠4 + ∠5 = 180°

It doesn't matter whether the star is regular or lopsided — the total is always 180°, the same as a single triangle.

Why it holds

Each tip is the apex of a small triangle whose base is a chord of the star. Chasing the base angles around the inner pentagon, every interior angle cancels in pairs, and what survives is exactly one triangle's worth of angle — 180°.

The bigger pattern

For a star polygon written {n/k} (connect every k-th of n points), the tip angles sum to 180°·(n − 2k). The five-pointed star is {5/2}, giving 180·(5 − 4) = 180°.

Keep it a star

The result needs a proper star: keep the five points in convex position (none tucked inside the others). The badge warns you if the shape stops being a clean pentagram.