Problem 580

Join $P$ to $R$. Since $PQRST$ is a regular pentagon, then $\angle PQR = \angle QRS = 108^{\circ}$. Since $PQ = QR$, then $\triangle PQR$ is isosceles with $\angle QPR = \angle QRP$. Since $\angle PQR = 108^{\circ}$, then $\angle PQR + \angle QPR + \angle QRP = 180^{\circ}$, $108^{\circ} + 2\angle QRP = 180^{\circ}$, $2\angle QRP = 72^{\circ}$, $\angle QRP = 36^{\circ}$. Since $\angle QRS = 108^{\circ}$, then $\angle PRS = \angle QRS - \angle QRP = 108^{\circ} - 36^{\circ} = 72^{\circ}$.