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

2017 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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2017 AMC 12A Problems/Problem 11

Contents

Problem

Solution 1

Solution 2 (fast with answer choices)

See Also