1. **State the problem:** We need to prove that triangles $\triangle TRS$ and $\triangle QRP$ are congruent given that $SR \cong PR$, $TQ$ is perpendicular to $TS$, and $TQ$ is perpendicular to $QP$. 2. **Given information:** - $SR \cong PR$ (given) - $TQ \perp TS$ (given) - $TQ \perp QP$ (given) 3. **Analyze the right angles:** Since $TQ$ is perpendicular to both $TS$ and $QP$, we know that: - $\angle TQS$ is a right angle ($90^\circ$) - $\angle TQP$ is a right angle ($90^\circ$) 4. **Identify the triangles and corresponding parts:** - Triangles: $\triangle TRS$ and $\triangle QRP$ - Sides: $SR \cong PR$ (given) - Angles: $\angle TQS$ and $\angle TQP$ are right angles and thus congruent. 5. **Check for shared side:** Both triangles share side $TR$ (common side). 6. **Apply the RHS (Right angle-Hypotenuse-Side) congruence criterion:** - Right angles: $\angle TQS \cong \angle TQP$ - Hypotenuse: $SR \cong PR$ - Side: $TR$ is common to both triangles Therefore, by RHS congruence, $\triangle TRS \cong \triangle QRP$. **Final answer:** $\boxed{\triangle TRS \cong \triangle QRP}$