Problem: An 'n-pointed star' is formed as follows: the sides of a convex polygon are numbered consecutively 1,2,β―,k,β―,n,nβ§5; for all n values of k, sides k and k+2 are non-parallel, sides n+1 and n+2 being respectively identical with sides 1 and 2; prolong the n pairs of sides numbered k and k+2 until they meet.
(A figure is shown for the case n=5 ).
Let S be the degree-sum of the interior angles at the n points of the star; then S equals:
Answer Choices:
A. 180
B. 360
C. 180(n+2)
D. 180(nβ2)
E. 180(nβ4)
Solution:
Let the measures of the angles at the n points be a1β,a2β,β¦,anβ and let Ξ±1β,Ξ±2β,β¦,Ξ±nβ be the measures of the interior angles of the polygon, with Ξ±nβ=Ξ±1β. We have a1β=180β(180βΞ±1β)β(180βΞ±2β)=Ξ±1β+Ξ±2ββ180, a2β=Ξ±2β+Ξ±3ββ180,β¦,anβ=Ξ±nβ1β+Ξ±nββ180. Summing, we have a1β+a2β+β¦+anβ=2(Ξ±1β+Ξ±2β+β¦+Ξ±nβ)βnβ
180
β΄S=2((nβ2)180)βnβ
180=180(nβ4)
