The image you uploaded contains two problems related to geometry in a circle. Both involve calculating angles associated with points on the circle and the center $M$. I'll break down the problems one by one.

Problem 1

You are asked to calculate the angles $\hat{A}_1$, $\hat{B}_1$, $\hat{C}_1$, $\hat{D}_1$, $\hat{M}_2$, and $\widehat{BCD}$ in a circle with center $M$, where $\hat{M}_1 = 100^\circ$.

Strategy:

• The center of the circle is given, and angles in the circle can be calculated using properties of cyclic quadrilaterals and inscribed angles.
• $\hat{M}_1 = 100^\circ$ is the angle subtended by arc $BC$ at the center. The inscribed angle on the arc opposite to it will be half the central angle.

Problem 2

You are asked to calculate the same set of angles in a similar configuration, but this time $\hat{A}_1 = 40^\circ$, which represents the inscribed angle at point $A$.

Strategy:

• Use the same properties of inscribed and central angles.
• The relation between the inscribed angles and the corresponding central angles is key.

General concepts:

• Inscribed Angle Theorem: The inscribed angle is half of the central angle subtended by the same arc.
• Cyclic Quadrilateral: Opposite angles in a cyclic quadrilateral add up to $180^\circ$.

Solution steps:

1. Central and Inscribed Angles: Identify which angles are subtended by specific arcs.
2. Symmetry and Angle Chasing: Use the given angles and symmetry of the circle to deduce other unknown angles.
3. Cyclic Quadrilateral Properties: Use opposite angle sums if necessary.

Would you like me to compute each angle step-by-step for both problems and explain the reasoning in detail?

Relative Questions:

1. What is the relationship between central and inscribed angles in a circle?
2. How does the Cyclic Quadrilateral Theorem help in angle calculations?
3. What are the key properties of angles subtended by the same arc?
4. How do you calculate angles when two diametrically opposite points are involved?
5. How can symmetry in geometric diagrams simplify angle calculations?

Tip: Always remember that an inscribed angle is half the size of the central angle subtended by the same arc.