Question

In the figure above, a shaded polygon which has equal sides and equal angles is partially covered with a sheet of blank paper. If $x+y={80}^o$, how many sides does the polygon have?

• A. Ten
• B. Nine
• C. Eight
• D. Seven
• E. Six

Answer 1

Answer: (B)

Nine

The key is to recognise 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 ${80}^o$, and a quadrilateral’s angles add up to ${360}^o$, you can conclude that the polygon angles add up to ${280}^o$, and therefore each angle in the polygon is ${140}^o$.

From there, if you know the $(n-2){180}^o$ thing, that’s a quick way to go:

$\frac{\left(\right(n-2\left){180}^o\right)}{n}={140}^o$
$\Rightarrow (n-2){180}^o=140n$
$\Rightarrow 180n-{360}^o=140n$
$\Rightarrow 40n={360}^o$

$\Rightarrow n=9$