CEMC Cayley, 2015

Solution 1

$\angle SRQ$ is an exterior angle of $\triangle PQR$.

Thus, $\angle SRQ = \angle RPQ + \angle PQR = 50^\circ + 90^\circ = 140^\circ$.

Therefore, $x^\circ = 140^\circ$ and so $x=140$.

Solution 2

The sum of the angles of $\triangle PQR$ is $180^\circ$, and so $$\angle PRQ = 180^\circ - \angle RPQ - \angle PQR = 180^\circ - 50^\circ - 90^\circ = 40^\circ$$ Since $\angle PRQ$ and $\angle SRQ$ are supplementary, then $x^\circ + 40^\circ = 180^\circ$, and so $x = 180 - 40 = 140$.