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In this problem, you are provided with the following information about a triangle \( \triangle PQR \):

### Given:
- \( PQ \cong QR \) (Side \( PQ \) is congruent to side \( QR \))
- \( QR = 14 \)
- \( PR = 14 \)

### Prove:
- \( PQ \cong PR \) (Side \( PQ \) is congruent to side \( PR \))

This proof involves showing that triangle \( PQR \) is isosceles by proving that two sides, \( PQ \) and \( PR \), are congruent.

### Two-Column Proof

| **Statements**                        | **Reasons**                             |
|---------------------------------------|-----------------------------------------|
| 1. \( PQ \cong QR \)                  | Given                                   |
| 2. \( QR = 14 \)                      | Given                                   |
| 3. \( PR = 14 \)                      | Given                                   |
| 4. \( PQ = PR \)                      | Transitive property of equality         |
| 5. \( PQ \cong PR \)                  | Definition of congruent segments (by length) |

In this proof:
- Step 1 acknowledges the given information that \( PQ \) is congruent to \( QR \).
- Step 2 and 3 use the given lengths of sides \( QR \) and \( PR \).
- Step 4 applies the transitive property (since \( PQ = QR = PR \), thus \( PQ = PR \)).
- Step 5 concludes that \( PQ \) is congruent to \( PR \) because congruent segments have equal lengths.

Would you like further details or explanations?

Here are 5 additional questions to explore this concept:
1. What other ways could you prove two sides of a triangle are congruent?
2. How do the properties of isosceles triangles relate to this proof?
3. Can we use the triangle inequality theorem to verify these side lengths?
4. What would change if the triangle were not isosceles?
5. How do definitions of congruent triangles help in geometry proofs?

**Tip:** When writing a proof, always double-check that every statement has a corresponding reason, ensuring logical consistency.

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

In this geometry problem, learn how to prove that sides PQ and PR of triangle PQR are congruent, given the lengths and properties of the triangle. The problem explores the transitive property of equality and the definition of congruent segments. Suitable for students in Grades 9-10.

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