1. **Problem statement:** We have a parallelogram PQRS and a rhombus RSTU with S on OT. Given $\angle QPS = 66^\circ$ and $\angle RTU = 37^\circ$, find $\angle PSO$. 2. **Key properties:** - In parallelogram PQRS, opposite sides are parallel and equal. - In rhombus RSTU, all sides are equal and opposite angles are equal. - Since S lies on OT, points O, T, S are collinear. 3. **Analyze parallelogram PQRS:** - $\angle QPS = 66^\circ$ is given. - Since PQRS is a parallelogram, $\angle PSQ = \angle QPS = 66^\circ$ because adjacent angles in parallelogram are supplementary and opposite angles are equal. 4. **Analyze rhombus RSTU:** - $\angle RTU = 37^\circ$ is given. - In rhombus, adjacent angles are supplementary, so $\angle RST = 180^\circ - 37^\circ = 143^\circ$. 5. **Find $\angle PSO$:** - Since S lies on OT, and O, T, S are collinear, $\angle PSO$ is the angle between PS and SO. - Note that $\angle PSO = 180^\circ - \angle RST = 180^\circ - 143^\circ = 37^\circ$. **Final answer:** $$\boxed{37^\circ}$$