Problem 192

Since the sum of the angles in a triangle is $180^\circ$, then $\angle QTR = 180^\circ - \angle TQR - \angle QRT = 180^\circ - 50^\circ - 90^\circ = 40^\circ$. Since opposite angles are equal, then $\angle STP = \angle QTR = 40^\circ$. Since the sum of the angles in $\triangle STP$ is $180^\circ$, then $$\begin{aligned} \angle PST + \angle SPT + \angle STP & = & 180^\circ \\ x^\circ + 110^\circ + 40^\circ & = & 180^\circ \\ x + 150 & = & 180 \\ x & = & 30\end{aligned}$$ Therefore, the value of $x$ is 30.