# 2017 AMC 12A Problems/Problem 11

## Problem

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? $\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163$

## Solution

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

## Solution 2 (fast with answer choices)

Because the sum of the interior angles is a multiple of 180, we know that the sum of the angles in a polygon is 0 mod 9. 2017 is congruent to 1 mod 9, so the answer has to be -1 mod 9. The only answer that is -1 mod 9 is 143. -harsha12345

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