Return of the Forgotten Angle!

Geometry Level 3

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of 2017 degrees. She then discovers that she forgot to include one angle.

What is the degree measure of the forgotten angle?

117 163 37 63 143

1 solution

Zach Abueg
Feb 7, 2017

We know that the sum of the interior angles of the polygon is a multiple of $180$ . Note that $\displaystyle \left\lceil\frac{2017}{180}\right\rceil = 12$ and $\displaystyle 180\cdot 12 = 2160$ , so the angle Claire forgot is $\displaystyle \equiv 2160-2017=143\mod 180$ . Since the polygon is convex, the angle is $\leq 180$ , so the answer is $\boxed{143}$ .

Nice question!

The question boils down to $(n-2) \times 180^\circ - m^\circ = 2017$ , where $m,n$ are positive integers with $n>3$ and $0<m<180$ (because it's a convex polygon).

This equation can also be interpreted as $2017/180 = \text{quotient } + \text{ remainder}$ . The value of $m$ can be found to be $- (2017 \bmod {180}) \equiv 143$ .

Pi Han Goh - 4 years, 4 months ago

Thanks! I agree, these are simpler ways to solve this problem.

Zach Abueg - 4 years, 3 months ago

Return of the Forgotten Angle!

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