\[\begin{align}  & \angle SQR=90{}^\circ \text{    }\!\!\{\!\!\text{ Tangent to a circle }\!\!\}\!\!\text{ } \\ & \angle QTS=\angle SQR=90{}^\circ \text{  }\!\!\{\!\!\text{ Alternate segment }\!\!\}\!\!\text{ } \\ & \angle TQS=\angle RSQ=50{}^\circ \text{  }\!\!\{\!\!\text{ Alternate }\angle \text{s }\!\!\}\!\!\text{ } \\ & \angle TQR=\angle RSQ+\angle SQR \\ & \angle TQR=50{}^\circ +90{}^\circ =140{}^\circ  \\ & \text{Consider }\vartriangle SQR \\ & \angle SRQ=180{}^\circ -\angle SQR-\angle RSQ \\ & \angle SRQ=180{}^\circ -90{}^\circ -50{}^\circ =40{}^\circ  \\ & \text{Consider the quadrilateral }TQRS \\ & \angle SRQ+\angle TQR+\angle STQ+\angle TSR=360{}^\circ  \\ & 40{}^\circ +140{}^\circ +90{}^\circ +\angle TSR=360{}^\circ  \\ & \angle TSR=90{}^\circ  \\ & \angle QST+\angle RSQ=\angle TSR \\ & \angle QST+50{}^\circ =90{}^\circ  \\ & \angle QST=40{}^\circ  \\\end{align}\]