Anonymous

free sample Collegeboard sat booklet 2011-2012, section 2 (math) question # 17. In the fig. above, a shaded polygon which has equal sides and equal angles is partially covered with a sheet of blank paper. If x + y = 80, how many sides does the polygon?

Yeah, this one is brutal. I actually adapted it for this old weekend challenge, the explanation for which you might want to check out since it contains some helpful diagrams.

The key is to recognize that the paper’s edge forms a quadrilateral with the shown part of the polygon. Since the two angles that aren’t part of the polygon add up to 80, and a quadrilateral’s angles add up to 360, you can conclude that the polygon angles add up to 280, and therefore each angle in the polygon is 140.

From there, if you know the (- 2)180 thing, that’s a quick way to go:

((- 2)180)/n = 140
(n - 2)180 = 140n
180n - 360 = 140n
40n = 360
n = 9

There’s also a clever solution to be found using triangles, which is how I explain it in the link above.