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.