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This image contains a geometric proof problem with the following information:

### Given:
1. \( \overline{NQ} \) bisects \( \angle MNO \).
2. \( \angle MNP \cong \angle QNR \).

### To Prove:
\( \angle PNO \cong \angle RNO \).

### Diagram:
The diagram shows a configuration with points \( M \), \( N \), \( O \), \( P \), \( Q \), and \( R \). \( \overline{NO} \) is a straight line, and \( \overline{NQ} \) bisects \( \angle MNO \), creating two angles. The angles \( \angle MNP \) and \( \angle QNR \) are marked as congruent.

### Proof Setup:
A two-column proof is started, and the first statement is given:
1. \( \overline{NQ} \) bisects \( \angle MNO \), and \( \angle MNP \cong \angle QNR \) (Reason: Given).

### Next Steps in Proof:
We need to show that \( \angle PNO \cong \angle RNO \). Here's the likely logic to follow:

### Statements:
2. \( \angle MNQ \cong \angle QNO \) (Definition of angle bisector).
3. \( \angle PNO \cong \angle RNO \) (This is the desired conclusion, possibly by Angle-Angle Similarity or the congruent triangles approach).

#### Would you like a more detailed explanation of how to complete this proof? Here are a few questions to expand your understanding:

1. Why does the angle bisector theorem apply in this proof?
2. How do we prove that triangles \( \triangle MNP \) and \( \triangle QNR \) are congruent?
3. Why does congruence of angles lead to the conclusion that \( \angle PNO \cong \angle RNO \)?
4. What other geometric principles can you use to support this proof?
5. How does the diagram help visualize the relationships between the angles?

**Tip:** Always mark known congruent angles and sides on the diagram when proving geometric theorems; it helps with visualizing the logic.

Given: NO bisects ∠MNO, ∠MNP ≅ ∠QNR. Prove: ∠PNO ≅ ∠RNO.

This problem involves proving that two angles (∠PNO and ∠RNO) are congruent, given that line NO bisects ∠MNO, and angles ∠MNP and ∠QNR are congruent. Explore the step-by-step geometric proof, utilizing the Angle Bisector Theorem and Congruent Angle relationships. Suitable for high school students.

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