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To determine if the two triangles, \( \triangle PNO \) and \( \triangle MNO \), are congruent, let's analyze the given information:

1. **Marked Angles**: Both triangles share angle \( \angle PNO \) and \( \angle MNO \), and they have equal markings. This indicates that these corresponding angles are congruent.
2. **Shared Side**: The line segment \( \overline{ON} \) is common to both triangles, meaning it is congruent to itself.

### Applying Congruence Postulates:
The Angle-Side-Angle (ASA) Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Here:
- **Angle \( \angle PNO \)** is congruent to **Angle \( \angle MNO \)**.
- **Side \( \overline{ON} \)** is congruent to itself (common side).
- **Angle \( \angle PON \)** is congruent to **Angle \( \angle MON \)**.

This satisfies the ASA Theorem.

### Conclusion:
The triangles \( \triangle PNO \) and \( \triangle MNO \) are congruent by the **ASA Theorem**.

### Answer:
- **Correct Option**: "The triangles are congruent by the ASA Theorem."

Would you like further details on the ASA Theorem or other congruence postulates?

### Related Questions:
1. What is the difference between the ASA and AAS postulates?
2. How would you apply the SAS Postulate to show triangle congruence?
3. Can triangles be congruent if only two angles are known?
4. What does it mean for two triangles to share a common side?
5. How would the analysis change if only one pair of angles was congruent?

### Tip:
Remember that the ASA Theorem requires that the congruent side must be between the two angles being compared!

Determine whether the triangles are congruent. If so, name the postulate or theorem that justifies your answer. If not, explain.

Learn how to determine the congruence of triangles using the ASA Theorem. This problem involves analyzing congruent angles and a shared side to verify triangle congruence, suitable for students in Grades 9-10.

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