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)== | ||
+ | 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 | ||
+ | |||
+ | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | ||
+ | https://youtu.be/4mVtT62THpA | ||
+ | |||
+ | ~Education, the Study of Everything | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2017|ab=A|num-b=10|num-a=12}} | {{AMC12 box|year=2017|ab=A|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:56, 10 June 2023
Contents
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
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
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 |
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