Question: Identify the proof to show that $\triangle PQS \cong \triangle RQS$, where $\angle QSP \cong \angle QSR$ are right angles, $S$ is the midpoint of $PR$, $PQ \cong QR$, and $QS$ bisects $\angle Q$
1. **State the problem:** We need to prove that $\triangle PQS \cong \triangle RQS$ given that $\angle QSP \cong \angle QSR$ are right angles, $S$ is the midpoint of $PR$, $PQ \cong QR$, and $QS$ bisects $\angle Q$.
2. **Identify given information and what to prove:**
- $\angle QSP \cong \angle QSR$ (both right angles) means $QS \perp PR$ at $S$.
- $S$ is midpoint of $PR$, so $PS = SR$.
- $PQ \cong QR$.
- $QS$ bisects $\angle Q$, so $\angle PQS \cong \angle RQS$.
3. **Use the RHS (Right angle-Hypotenuse-Side) congruence criterion:**
- Both triangles have a right angle at $S$.
- Hypotenuses $PQ$ and $QR$ are congruent.
- One leg $PS$ equals $SR$ since $S$ is midpoint.
4. **Write the congruence statement:**
$$\triangle PQS \cong \triangle RQS$$
by RHS congruence.
5. **Explain:**
Since $S$ is midpoint, $PS = SR$.
Both triangles share side $QS$.
Both have right angles at $S$.
Hypotenuses $PQ$ and $QR$ are equal.
Therefore, by RHS, the triangles are congruent.
**Final answer:**
$$\triangle PQS \cong \triangle RQS$$