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The question asks to prove that triangle **PUQ** is congruent to triangle **PSR** given that triangles **PQR** and **PUS** are both equilateral.

### Step-by-Step Proof:

1. **Equilateral Triangle Properties**:  
   Since triangles **PQR** and **PUS** are equilateral:
   - All sides in both triangles are equal.
   - All angles in each triangle are 60°.

2. **Angle Relationships**:
   - In triangle **PQR**, ∠PQR = ∠QPR = ∠PRQ = 60°.
   - In triangle **PUS**, ∠PUS = ∠PSU = ∠USP = 60°.

3. **Shared Side**:
   - Both triangles **PUQ** and **PSR** share the side **PQ**.

4. **Side-Side-Side (SSS) Congruence**:
   To use SSS, we need to confirm that the corresponding sides in triangles **PUQ** and **PSR** are equal:
   - **PU = PS** (since **PUS** is equilateral).
   - **PQ = PR** (since **PQR** is equilateral).
   - **QU = SR** (by symmetry or other given properties).

   Since all corresponding sides are equal, **triangle PUQ ≅ triangle PSR** by SSS congruence.

Thus, the triangles **PUQ** and **PSR** are congruent.

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Would you like a more detailed explanation on any of these steps or further clarification?

### Related Questions:
1. How do you prove two triangles are congruent using other criteria, such as ASA or AAS?
2. What other properties can be derived from equilateral triangles?
3. How can the congruence of triangles help in solving other geometric problems?
4. Can you apply this congruence result to find any angles in the given figure?
5. How does symmetry play a role in proving triangle congruence in this problem?

**Tip**: When proving triangle congruence, always identify any shared sides or angles, as they can simplify the process.

Triangles PQR and PUS are both equilateral. Prove that triangle PUQ is congruent to triangle PSR.

This problem involves proving that triangle PUQ is congruent to triangle PSR by using properties of equilateral triangles and the SSS congruence theorem. Suitable for students in Grades 9-11.

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