Problem:
Part of an " n-pointed regular star" is shown. It is a simple closed polygon in which all 2n edges are congruent, angles A1​,A2​, …,An​ are congruent and angles B1​,B2​,…,Bn​ are congruent. If the acute angle at A1​ is 10∘ less than the acute angle at B1​, then n=
Answer Choices:
A. 12
B. 18
C. 24
D. 36
E. 60
Solution:
Partition the n-pointed regular star into the regular n-gon B1​B2​⋯Bn​ and n triangles congruent to △B1​A2​B2​, and note that the sum of the star's interior angles is
(n−2)180∘+n180∘=(2n−2)180∘.
Since the interior angles of the star consist of n angles congruent to A1​ and n angles congruent to 360∘−B1​,
(2n−2)180∘=n∠A1​+n(360∘−∠B1​), or n(∠B1​−∠A1​)=2⋅180∘.
Since ∠B1​−∠A1​=10∘,n=36.
Note. In general, the sum of the interior angles of any N-sided simple closed polygon, convex or not, is (N−2)180∘.
