The angles at the points of a star

Question

Take any five points in convex position and join them in the order 1 to 3 to 5 to 2 to 4 and back to 1, producing a five-pointed star drawn in one stroke (a pentagram, not necessarily regular).

What is the sum of the five angles at the star's points? Prove that it is the same for every such star, and generalise to seven-pointed stars, drawn by joining every second point and by joining every third point.

Accepted Answer

The five point angles always add to 180 degrees. The inner pentagon. The star's lines cross to form a small pentagon in the middle. Each point of the star is a triangle sitting on one side of that inner pentagon; the triangle's two base angles are exterior angles of the inner pentagon (they are the angles between a side and the extension of a neighbouring side). Add up the triangles. Each point angle equals 180 degrees minus the two base angles of its triangle. Summing over the five points: sum of point angles = 5 × 180 minus (sum of all ten base angles). Every exterior angle of the inner pen…

The angles at the points of a star

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