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

The triangles are not necessarily congruent.  Not enough information provided.

The triangles are congruent by the ASA Theorem.

The triangles are congruent by the AAS Postulate.

The triangles in the diagram appear to be \( \triangle PNO \) and \( \triangle MNO \), with \( NO \) acting as a common side between them. Additionally, we see that:

1. \( \angle PNO = \angle MNO \), which are vertical angles and therefore congruent.
2. \( \angle NPO = \angle NMO \), marked by the red arcs, indicating they are congruent as well.

These conditions suggest that the two triangles have two angles and the included side (the side between the two angles), \( NO \), that are congruent.

Since we have two angles and the included side congruent in both triangles, this satisfies the **ASA (Angle-Side-Angle) Theorem** for triangle congruence.

### Answer:
The triangles are congruent by the **ASA Theorem**.

Would you like a detailed breakdown or have any questions on this explanation?

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### Follow-Up Questions:
1. How does the ASA Theorem differ from other congruence postulates, such as SAS or SSS?
2. What is the significance of the included side in the ASA theorem?
3. Can the AAS Postulate be applied in cases where the side is not included between the angles? How does that affect congruence?
4. How would you prove that two triangles are congruent using the SSS Postulate?
5. Are there any cases where two triangles cannot be proven congruent even if some sides or angles are equal?

### Tip:
When proving congruence in triangles, identify pairs of congruent angles and sides carefully. Different theorems apply based on whether the congruent side is included between two angles or not.

Determine whether the triangles are congruent. If so, name the postulate or theorem that justifies your answer. If not, explain. The triangles are congruent by the SAS Postulate. The triangles are not necessarily congruent. Not enough information provided. The triangles are congruent by the ASA Theorem. The triangles are congruent by the AAS Postulate.

Explore how to determine if two triangles are congruent using the ASA, SAS, or AAS postulates. This problem focuses on identifying angle relationships and applying congruence theorems in geometry, ideal for students in Grades 8-10.

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