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Problem 266

Geometry Difficulty 2.1 Multiple choice CEMC Fermat · Canada · 2013

In the diagram, $PQRS$ is a square and $PTQ$ is an equilateral triangle.

What is the measure of $\angle TPR$ ?

Pick one

Official solution

Each interior angle in a square is $90^\circ$ . In particular, $\angle SPQ = 90^\circ$ .

Each interior angle in an equilateral triangle is $60^\circ$ . In particular, $\angle TPQ = 60^\circ$ .

$PR$ is a diagonal of square $PQRS$ . Thus, it bisects angle $\angle SPQ$ , with $\angle SPR = \angle RPQ = 45^\circ$ .

Therefore, $\angle TPR = \angle TPQ + \angle QPR = 60^\circ + 45^\circ = 105^\circ$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.