Library / CEMC Fermat /309 of 566

CEMC Fermat, 2013

Geometry Difficulty 2.1 Junior Find the answer Canada

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

What is the measure of $\angle TPR$ ?

Pick one

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

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