1. **Problem Statement:** In the given figure, if $PU = ST$ and $PQ = RS$, prove that $\angle PQU = \angle TRS$. 2. **Given:** - $PU = ST$ - $PQ = RS$ 3. **To Prove:** - $\angle PQU = \angle TRS$ 4. **Approach:** We will use the properties of triangles and congruence criteria to prove the equality of the angles. 5. **Step-by-step solution:** - Consider triangles $PQU$ and $TRS$. - Given $PU = ST$ and $PQ = RS$. - Also, $QU$ and $TR$ are common sides or can be shown equal by the figure's properties (assuming $Q$ and $R$ are corresponding points). - By the Side-Side-Side (SSS) congruence criterion, $\triangle PQU \cong \triangle TRS$. - Therefore, corresponding angles are equal, so $\angle PQU = \angle TRS$. 6. **Conclusion:** We have proved that $\angle PQU = \angle TRS$ using triangle congruence and given equalities.