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We are tasked with identifying which of the given circle diagrams contain two angles subtended by the same arc. 

### Key Concept:
For angles to be subtended by the same arc, they must:
1. Lie on the circumference of the circle.
2. Share the same two endpoints on the circle.

Let’s analyze each diagram:

### Diagram A:
- The two angles seem to share the same endpoints on the circumference, which are the two ends of a common chord. Both angles are on the circumference, subtended by the same arc.
- **Conclusion: Yes, the two angles are subtended by the same arc.**

### Diagram B:
- In this case, the two angles appear to be on different arcs. One angle is subtended by one arc, and the other is subtended by another arc.
- **Conclusion: No, the two angles are not subtended by the same arc.**

### Diagram C:
- This figure shows two angles that share the same arc endpoints and are on the circumference. Both angles are subtended by the same arc.
- **Conclusion: Yes, the two angles are subtended by the same arc.**

### Diagram D:
- While there are multiple angles, they do not seem to be subtended by the same arc. The two angles are subtended by different chords and arcs.
- **Conclusion: No, the two angles are not subtended by the same arc.**

### Final Answer:
- **A and C** have angles subtended by the same arc.

Would you like further clarification, or details on any specific part? Here are some questions that expand on this:

1. What is the formal definition of a subtended angle in a circle?
2. How do you identify angles subtended by the same arc?
3. Why do diagrams B and D fail to meet the criteria?
4. Can angles subtended by different arcs ever be congruent?
5. How does the "angle at the center" theorem relate to subtended angles?

**Tip:** Remember that angles subtended by the same arc are always equal!

For each of the following circle figures, are the two angles subtended by the same arc?

Learn how to identify angles subtended by the same arc in circle diagrams. This problem covers key geometric concepts such as the angle subtended by the same arc theorem, ideal for students in Grades 9-10.

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