¶ 2009 AMC 10B Problem 24
Problem:
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with trapezoids, let be the angle measure in degrees of the larger interior angle of the trapezoid. What is ?
Answer Choices:
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Solution:
Add a symmetric arch to the given arch to create a closed loop of trapezoids. Consider the regular -sided polygon created by the interior of the completed loop. Each interior angle of a regular -gon measures
Then , so .
Extend two sides of a trapezoid until they meet at the center of the arch, as shown. Then is isosceles and by symmetry , and . The requested angle is supplementary to , so .
The problems on this page are the property of the MAA's American Mathematics Competitions