To determine which theorem could be used to prove that $\triangle PQR \cong \triangle TSR$ based on the given information ($QR \cong SR$ and $PR \cong TR$):

Step-by-step Explanation:

1. Given Information:

   • $QR \cong SR$ (two sides are congruent).
   • $PR \cong TR$ (two sides are congruent).

2. Shared Angle:

   • Both triangles share $\angle PRQ = \angle SRT$ at $R$, making this angle congruent.

3. Congruence Criterion:

   • Two sides of the triangles are congruent ($QR \cong SR$, $PR \cong TR$), and the angle between those sides ($\angle PRQ$) is also congruent.

Thus, the correct theorem is Side-Angle-Side (SAS).

Final Answer:

The theorem that could be used to prove $\triangle PQR \cong \triangle TSR$ is: Side-Angle-Side (SAS).

Would you like further clarification or details about the SAS theorem or related concepts?

Expanding Questions:

1. Can you explain why the Angle-Side-Angle (ASA) theorem doesn’t apply here?
2. What are the conditions under which the Side-Side-Side (SSS) theorem can be used?
3. How does the SAS theorem differ from the Side-Side-Angle (SSA) method?
4. Can you identify a practical use case for triangle congruence in real-world applications?
5. How would the proof change if the given congruent parts were sides and angles not included in this configuration?

Tip: When solving problems with congruence, always ensure the specified angles are included between the congruent sides to apply SAS correctly!