Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1982 AHSME Problems/Problem 6"

(created page w/ solution & categorization)
 
m (Add title)
 
Line 1: Line 1:
 +
== Problem ==
 +
 
The sum of all but one of the interior angles of a convex polygon equals <math>2570^\circ</math>. The remaining angle is
 
The sum of all but one of the interior angles of a convex polygon equals <math>2570^\circ</math>. The remaining angle is
  

Latest revision as of 12:27, 16 July 2024

Problem

The sum of all but one of the interior angles of a convex polygon equals $2570^\circ$. The remaining angle is

$\text{(A)} \ 90^\circ \qquad \text{(B)} \ 105^\circ \qquad \text{(C)} \ 120^\circ \qquad \text{(D)}\ 130^\circ\qquad \text{(E)}\ 144^\circ$

Solution

Note that the sum of the interior angles of a convex polygon of $n$ sides is $180(n-2)^\circ$, and each interior angle belongs to $[0, 180^\circ)$. Therefore, we must have $n - 2 = \lfloor \frac{2570^\circ}{180^\circ} \rfloor = 15$. Then the missing angle must be $180*15^\circ - 2570^\circ = 130^\circ$, so our answer is $\boxed{\text{(D)}}$ and we are done.

See also

1982 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1982_AHSME_Problems/Problem_6&oldid=222786"