https://file.aiquickdraw.com/mathbot/file/1729448264092xd8sixzm.jpg

To solve this, we will use the information from the diagram and apply geometry concepts.

We are tasked with calculating the angles \( \angle AMB \) and \( \angle APB \). Here's the breakdown:

### Step 1: Analyzing \( \angle AMB \)

The angle \( \angle AMB \) is a central angle in the circle. Given:
- \( \angle AMP = 60^\circ \)
- \( \angle AMB = 2 \times \angle AMP \)

This is because the angle at the center (subtended by the arc) is double the angle subtended by the same arc at the circumference (inscribed angle theorem).

Thus:
\[
\angle AMB = 2 \times 60^\circ = 120^\circ
\]

### Step 2: Analyzing \( \angle APB \)

The angle \( \angle APB \) is an inscribed angle, and it is subtended by the same arc as the central angle \( \angle AMB \). From the inscribed angle theorem, we know:
\[
\angle APB = \frac{1}{2} \times \angle AMB
\]

Substituting \( \angle AMB = 120^\circ \):
\[
\angle APB = \frac{1}{2} \times 120^\circ = 60^\circ
\]

### Conclusion:
- \( \angle AMB = 120^\circ \)
- \( \angle APB = 60^\circ \)

Would you like more details or explanations on any part of this solution?

Here are five related questions you might find helpful:
1. How does the inscribed angle theorem work in circles?
2. Can the angles \( \angle AMB \) and \( \angle APB \) ever be equal?
3. What happens to the angle \( \angle AMB \) if \( \angle AMP \) changes?
4. How can we prove that \( \angle AMB = 2 \times \angle AMP \) using circle geometry?
5. What role does the arc play in determining the size of the angles in the circle?

**Tip:** In a circle, the angle subtended by an arc at the center is always twice the angle subtended by the same arc at the circumference.

Bereken ∠AMB en ∠APB met de gegevens in de figuur.

Learn how to calculate the angles ∠AMB and ∠APB using circle geometry and theorems. We explore the relationship between central and inscribed angles, applying key concepts like the Central Angle Theorem and the Inscribed Angle Theorem. Suitable for students in Grades 9-12.

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