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This is a two-column proof from geometry, and the goal is to prove that \( \overline{QR} \cong \overline{NO} \).

### Let's fill in the missing steps in the proof based on the information provided:

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**Given:**  
- \( \overline{MN} \cong \overline{PQ} \)  
- \( \overline{MO} \cong \overline{PR} \)

**To Prove:**  
- \( \overline{QR} \cong \overline{NO} \)

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### Proof:

| **Statements** | **Reasons** |
|:---|:---|
| 1. \( \overline{MN} \cong \overline{PQ} \) and \( \overline{MO} \cong \overline{PR} \) | 1. Given |
| 2. \( PQ + QR = PR \) and \( MN + NO = MO \) | 2. Segment Addition Postulate |
| 3. \( PR = PQ + QR \) | 3. Definition of Congruent Segments |
| 4. \( MO = MN + NO \) | 4. Definition of Congruent Segments |
| 5. \( MN + NO = PQ + QR \) | 5. Substitution Property of Equality |
| 6. \( MN \cong PQ \) | 6. Given |
| 7. \( NO = QR \) | 7. Substitution Property of Equality |
| 8. \( NO \cong QR \) | 8. Symmetric Property of Equality |
| 9. \( \overline{QR} \cong \overline{NO} \) | 9. Definition of Congruent Segments |

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This concludes the proof that \( \overline{QR} \cong \overline{NO} \).

Do you have any questions or would you like more details on any of these steps? Here are 5 questions you can explore further:

1. What is the Segment Addition Postulate, and how does it work?
2. Why is the Substitution Property of Equality used in step 5?
3. How is the Symmetric Property of Equality applied in geometry proofs?
4. What is the difference between congruent segments and equal lengths?
5. How can this proof method be applied to other geometric figures?

### Tip:
When working on geometric proofs, always look for ways to substitute or manipulate the given information using postulates and properties to make the necessary connections.

Prove that QR ≅ NO given MN ≅ PQ and MO ≅ PR.

Learn how to prove that line segments QR and NO are congruent using given segment relations MN ≅ PQ and MO ≅ PR. This problem uses the Segment Addition Postulate and properties of equality, suitable for students in Grades 8-10.

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