Answer:
116
Step-by-step explanation:
Since the two angles at $R$ add to $180^\circ,$ then $\angle QRT + 128^\circ = 180^\circ,$ so $\angle QRT = 52^\circ.$
Since $PT$ and $QR$ are parallel, then $\angle PTR$ and $\angle QRT$ are supplementary, so $2x^\circ + 52^\circ = 180^\circ$ or $2x^\circ = 128^\circ$ or $x = 64.$
Therefore, three of the angles of quadrilateral $PQRT$ are $64^\circ,$ $128^\circ,$ and $52^\circ.$
Since the angles in a quadrilateral add to $360^\circ,$ then $\angle PQR = 360^\circ - 64^\circ - 128^\circ - 52^\circ = \boxed{116} \text{ degrees}$.
