# 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 1

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 congruent to $-1 \mod 9$ is $143$. -harsha12345

## See Also

 2017 AMC 12A (Problems • Answer Key • Resources) Preceded byProblem 10 Followed byProblem 12 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. Invalid username
Login to AoPS