This problem involves proving that triangles $\triangle PQR$ and $\triangle STR$ are congruent. Let’s break down the information step-by-step and work through the proof.

Given:

1. $\angle PQR$ is a right angle.
2. $\angle STR$ is a right angle.
3. $QR \cong TR$ (the two segments are congruent).

To Prove:

• $\triangle PQR \cong \triangle STR$

Steps for Proof (using the SAS Congruence Theorem):

We will use the Side-Angle-Side (SAS) congruence theorem, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

1. Step 1: $\angle PQR$ and $\angle STR$ are both right angles.

   • Reason: Given in the problem.

2. Step 2: $QR \cong TR$

   • Reason: Given in the problem.

3. Step 3: $PR \cong SR$ (the vertical line segment between the two right triangles).

   • Reason: Common side (reflexive property).

4. Step 4: $\triangle PQR \cong \triangle STR$

   • Reason: By SAS (Side-Angle-Side) theorem. We have two sides and the included right angle congruent.

Would you like a deeper explanation of each step or have any questions?

Here are 5 relative questions that expand on this concept:

1. What other congruence theorems can be used to prove triangle congruence?
2. How would the proof change if the right angles were not given?
3. Can the triangles still be congruent if one of the sides wasn't given as congruent?
4. What is the importance of the reflexive property in geometric proofs?
5. How would you approach proving triangles similar rather than congruent?

Tip: Always make sure to check if you can apply the reflexive property, as it's commonly used when two triangles share a side.