Let's work through this proof together. You are asked to complete the proof that $\triangle PQR \cong \triangle RSP$ based on the given conditions and the statements already laid out.

Given Information:

• $QR \perp PQ$
• $PQ \parallel RS$
• $PS \perp RS$

Statements and Justifications:

1. $QR \perp PQ$
   Reason: Given. It establishes that $\angle QRP$ is a right angle.

2. $PQ \parallel RS$
   Reason: Given. This means that corresponding angles formed by the transversal will be congruent.

3. $PS \perp RS$
   Reason: Given. It establishes that $\angle PRS$ is a right angle.

4. $\angle Q \cong \angle S$
   Reason: Corresponding angles formed by the parallel lines $PQ \parallel RS$ and the transversal $PR$.

5. $\angle QPR \cong \angle PRS$
   Reason: Both angles are right angles, which makes them congruent by definition of perpendicular lines.

6. $PR = PR$
   Reason: Reflexive property of congruence. $PR$ is common to both triangles.

7. $\triangle PQR \cong \triangle RSP$
   Reason: Angle-Angle-Side (AAS) Congruence Theorem. Two angles and the included side are congruent.

This completes the proof!

Follow-up Questions:

1. Why is the AAS (Angle-Angle-Side) theorem applicable here?
2. How does the property of perpendicular lines help in proving triangle congruence?
3. Can we apply the SAS (Side-Angle-Side) theorem instead in this proof? Why or why not?
4. What other pairs of triangles in the diagram might be congruent using similar reasoning?
5. How do parallel lines contribute to the congruence of triangles in geometry?

Tip: When dealing with triangle congruence, always check if shared sides or angles can be used with the Reflexive Property to simplify your reasoning.