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The given problem involves two triangles, \( \triangle PQR \) and \( \triangle TSR \), and we are tasked with proving that \( \triangle PQR \cong \triangle TSR \).

### Given:
- \( \overline{QR} \cong \overline{SR} \) (side congruence)
- \( \overline{PR} \cong \overline{TR} \) (side congruence)

### To Prove:
- \( \triangle PQR \cong \triangle TSR \)

### Proof:
We can prove the congruence of \( \triangle PQR \) and \( \triangle TSR \) using the **SSS (Side-Side-Side) Congruence Theorem**. Here's how:

1. **First Pair of Congruent Sides**: From the given, \( \overline{QR} \cong \overline{SR} \). This means the side connecting points \( Q \) and \( R \) in \( \triangle PQR \) is congruent to the side connecting points \( S \) and \( R \) in \( \triangle TSR \).

2. **Second Pair of Congruent Sides**: Also given, \( \overline{PR} \cong \overline{TR} \). This tells us that the side \( PR \) in \( \triangle PQR \) is congruent to side \( TR \) in \( \triangle TSR \).

3. **Third Pair of Congruent Sides**: Notice that \( \overline{QR} \) and \( \overline{SR} \) both share point \( R \), making \( \overline{RS} \) a common side to both triangles. Since this side is the same in both triangles, we have \( \overline{RS} \cong \overline{RS} \) (the **Reflexive Property** of equality).

Since all three pairs of corresponding sides of \( \triangle PQR \) and \( \triangle TSR \) are congruent, we conclude that:

\[
\triangle PQR \cong \triangle TSR \quad \text{(by SSS Congruence Theorem)}
\]

Would you like more details or have any questions?

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Here are 5 additional questions related to this proof:

1. How does the SSS Congruence Theorem work in triangle congruence?
2. What are other triangle congruence theorems besides SSS?
3. How can you prove congruence using angle-side-angle (ASA) instead?
4. What is the reflexive property, and how is it used in proofs?
5. Can congruence be established if only two sides and one angle are known?

**Tip**: In geometry, always check for shared sides or angles between triangles, as these can often be used to simplify proofs using the reflexive property.

Given that QR ≅ SR and PR ≅ TR, prove that ΔPQR ≅ ΔTSR.

This geometry problem focuses on proving the congruence of triangles PQR and TSR using the SSS (Side-Side-Side) Congruence Theorem. It involves the congruence of sides QR ≅ SR and PR ≅ TR, suitable for students in Grades 9-11.

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