2009 AMC 10B Problem 24

Question

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

9

trapezoids, let

x

be the angle measure in degrees of the larger interior angle of the trapezoid. What is

x?

100

102

104

106

108

Answer: A

Concepts:regular polygon angle sum symmetry

Difficulty rating: 1600

Small hint:

Reflect the arch to close it into a full loop of

18

trapezoids

Big hint:

The inner vertices form a regular

18

-gon; use its interior angle

Solution:

Adding a mirror image completes the arch into a symmetric closed loop of

18

trapezoids. Their inner edges form a regular

18

-gon, each interior angle of which is

\dfrac{(18-2)\cdot180^\circ}{18}=160^\circ.

At each inner vertex, two of the trapezoids’ larger angles

x

meet the

160^\circ

angle around a full turn:

x+x+160^\circ=360^\circ,

so

x=100.

Thus, the correct answer is A.