An American High School Mathematics Examination Question

This Demonstration is an attempt to produce multiple-choice questions such as those posed on various examinations like the AHSME (American High School Mathematics Examination) and the AIME (American Invitational Mathematics Examination), or at the Australian mathematics competition and the European Mathematical Kangaroo. Here is a generalization of question 30 on the 1983 AHSME.

In and , we have (in the order given) the condition angle-side-side. By the "ASS theorem", the angles and are either equal or supplementary. Since these triangles are not congruent (), we must have that and are supplementary. From we compute

.

Thus, and

.

Reference

[1] G. Berzsenyi and S. B. Maurer, eds., The Contest Problem Book V, Washington, DC: Mathematical Association of America, 1998.