smc

Question

FIGURE Part of an "n-pointed regular star" is shown. It is a simple closed polygon in which all 2n edges are congruent, angles A_1,A_2,,A_n are congruent, and angles B_1,B_2,,B_n are congruent. If the acute angle at A_1 is 10^ less than the acute angle at B_1, then n=

Accepted Answer

If we sum up the angles to obtain 360, we can see that the B angles add to the sum and the A angles subtract from the sum (an easy way of looking at this is by using the opposing angle theorem: if A[n] = B[n] than their total contribution is 0). Thus we have B[1] + B[2] + ... + B[n] - A[1] - A[2] - ... A[n] = 360. But every pair of A,B has a total 'angle contribution' of 10, thus there are 36 36 pairs of A,B.