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The given problem shows a geometric figure, specifically two triangles, with the following information provided:

### Given:
1. \( PV \cong PR \) (Line segment \( PV \) is congruent to line segment \( PR \)).
2. \( VT \cong RS \) (Line segment \( VT \) is congruent to line segment \( RS \)).

### Conclusion:
- The problem asks to conclude that \( PT \cong PS \) (Line segment \( PT \) is congruent to line segment \( PS \)).

### Reasoning:
We can apply the **SAS (Side-Angle-Side) Congruence Theorem** to prove that the triangles involved are congruent:

1. In triangles \( PVT \) and \( PRS \):
   - \( PV \cong PR \) (Given).
   - \( VT \cong RS \) (Given).
   - \( \angle P \) is common to both triangles \( PVT \) and \( PRS \).
   
Thus, by the **SAS Congruence Theorem**, we conclude that triangles \( PVT \) and \( PRS \) are congruent.

### Conclusion:
Since the two triangles \( PVT \) and \( PRS \) are congruent, the corresponding parts of congruent triangles are congruent (CPCTC), which means that \( PT \cong PS \), as required.

Would you like further clarification or a breakdown of the theorem applied? Here are five related questions for further practice:

1. What is the **SAS Congruence Theorem**, and how does it apply to triangle congruence?
2. Can the **ASA Congruence Theorem** also be used in this scenario? Why or why not?
3. What does **CPCTC** stand for, and how is it used in congruence proofs?
4. Could this problem be solved using a different triangle congruence theorem? Which one?
5. How does the concept of congruence apply in coordinate geometry?

**Tip**: When proving congruence, always look for common angles or sides between the triangles to simplify your proof.

Given: PV ≅ PR, VT ≅ RS. Conclusion: PT ≅ PS.

In this geometry problem, we prove that PT ≅ PS using the SAS Congruence Theorem. Given that PV ≅ PR and VT ≅ RS, and that both triangles share the same angle at P, we apply SAS and CPCTC to conclude that PT ≅ PS. Suitable for students in Grades 8-10.

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