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sum of assigned angles is multiple of 720 then odd number of self-intersections

Source: Ukrainian TST 1999 p7

February 13, 2020
combinatoricscombinatorial geometryangles

Problem Statement

Let P1P2...PnP_1P_2...P_n be an oriented closed polygonal line with no three segments passing through a single point. Each point PiP_i is assinged the angle 180o−∠Pi−1PiPi+1≥0180^o - \angle P_{i-1}P_iP_{i+1} \ge 0 if Pi+1P_{i+1} lies on the left from the ray Pi−1PiP_{i-1}P_i, and the angle −(180o−∠Pi−1PiPi+1)<0-(180^o -\angle P_{i-1}P_iP_{i+1}) < 0 if Pi+1P_{i+1} lies on the right. Prove that if the sum of all the assigned angles is a multiple of 720o720^o, then the number of self-intersections of the polygonal line is odd
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