Difference between revisions of "2017 AMC 12A Problems/Problem 11"
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<math>\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163</math> | <math>\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163</math> | ||
| − | ==Solution== | + | ==Solution 1== |
We know that the sum of the interior angles of the polygon is a multiple of <math>180</math>. Note that <math>\left\lceil\frac{2017}{180}\right\rceil = 12</math> and <math>180\cdot 12 = 2160</math>, so the angle Claire forgot is <math>\equiv 2160-2017=143\mod 180</math>. Since the polygon is convex, the angle is <math>\leq 180</math>, so the answer is <math>\boxed{(D)\ =\ 143}</math>. | We know that the sum of the interior angles of the polygon is a multiple of <math>180</math>. Note that <math>\left\lceil\frac{2017}{180}\right\rceil = 12</math> and <math>180\cdot 12 = 2160</math>, so the angle Claire forgot is <math>\equiv 2160-2017=143\mod 180</math>. Since the polygon is convex, the angle is <math>\leq 180</math>, so the answer is <math>\boxed{(D)\ =\ 143}</math>. | ||
==Solution 2 (fast with answer choices)== | ==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. | + | Because the sum of the interior angles is a multiple of <math>180</math>, we know that the sum of the angles in a polygon is <math>0 \mod 9</math>. <math>2017</math> is congruent to <math>1 \mod 9</math>, so the answer has to be <math>-1 \mod 9</math>. The only answer that is congruent to <math>-1 \mod 9</math> is <math>143</math>. |
-harsha12345 | -harsha12345 | ||
Revision as of 13:19, 19 December 2020
Problem
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of 
. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
Solution 1
We know that the sum of the interior angles of the polygon is a multiple of 
. Note that 
and 
, so the angle Claire forgot is 
. Since the polygon is convex, the angle is 
, so the answer is 
.
Solution 2 (fast with answer choices)
Because the sum of the interior angles is a multiple of , we know that the sum of the angles in a polygon is 
. is congruent to 
, so the answer has to be 
. The only answer that is congruent to is 
.
-harsha12345
See Also
| 2017 AMC 12A (Problems • Answer Key • Resources) | |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
