Pr and ps are tangent to center q. find the measurement of angle q

Segment PQ bisects the angle P, which means angle QPR has a measure of 21 degrees (which follows from the fact that triangles PQR and PQS are congruent and PR and PS are tangent to the circle).

This then means angle RQP has measure 90 - 21 = 69 degrees.

Angle Q is twice this, so it has measure 138 degrees.