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Geometry Difficulty 3.4 AMC 10/12 Find the answer Canada

In the diagram, $\triangle PQR$ is right-angled at $Q$ and has $\angle QPR = 54^\circ$ . Also, point $S$ lies on $PQ$ such that $\angle PRS = \angle QRS$ .What is the measure of $\angle RSQ$ ?

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Solution

Since $\triangle PQR$ is right-angled at $Q$ , then $\angle PRQ = 90^\circ - \angle QPR = 90^\circ - 54^\circ = 36^\circ$ . Since $\angle PRS = \angle QRS$ , then $\angle QRS = \frac{1}{2}\angle PRQ = \frac{1}{2}(36^\circ) = 18^\circ$ . Since $\triangle RQS$ is right-angled at $Q$ , then $\angle RSQ = 90^\circ - \angle QRS = 90^\circ - 18^\circ = 72^\circ$ .

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