https://file.aiquickdraw.com/mathbot/file/1726348443421foicj860.jpg

The problem in the image involves a triangle \( \triangle PQR \) with the following information:

**Given:**
- \( PQ \cong QR \)
- \( QR = 14 \)
- \( PR = 14 \)

**To Prove:**
- \( PQ \cong PR \)

### Proof:

1. **Given:**
   - \( PQ \cong QR \) (it is stated in the problem)
   - \( QR = 14 \)
   - \( PR = 14 \)

2. Since \( PR = 14 \) and \( QR = 14 \), we can conclude that \( PR \cong QR \) (they have the same length).

3. Also, we know from the given that \( PQ \cong QR \). Therefore, we have \( PQ \cong PR \) by the transitive property of equality (if two segments are both congruent to a third segment, they are congruent to each other).

### Conclusion:
Thus, \( PQ \cong PR \), as required.

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Would you like further details or have any questions? Here are some follow-up questions:

1. How does the transitive property of equality work in geometry?
2. Can you explain why two sides of a triangle being equal make it isosceles?
3. How does the congruence of sides affect the angles in a triangle?
4. What is the difference between congruent and equal in geometry?
5. How does triangle congruence relate to similarity?

**Tip:** When working with geometric proofs, always keep properties like symmetry and transitivity in mind, as they help in proving congruence efficiently.

Given: PQ ≅ QR, QR = 14, PR = 14. Prove: PQ ≅ PR

This problem involves proving that side PQ is congruent to side PR in triangle PQR using the given congruences and the transitive property of equality. Suitable for students in Grades 8-10 learning about triangle congruence and geometric proofs.

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