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?

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.