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

See Also

2017 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 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

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