HMMT November

Geometry Difficulty 4.5 AIME Find the answer

Let $ABCD$ be a trapezoid such that $AB \parallel CD, \angle BAC=25^{\circ}, \angle ABC=125^{\circ}$ , and $AB+AD=CD$ . Compute $\angle ADC$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Construct the parallelogram $ABED$ . From the condition $AB+AD=CD$ , we get that $EC=AD=EB$ . Thus, $\angle ADC=\angle BEC=180^{\circ}-2 \angle BCE=180^{\circ}-2 \cdot 55^{\circ}=70^{\circ}$ .

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