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 $x?$

$100$

$102$

$104$

$106$

$108$

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.

2009 AMC 10B Problem 24

Solution:

Problem 24 in Other Years