Let's work through each diagram in the image and find the angles marked with letters.

Key Concepts:

• Angles subtended by the same arc in a circle are equal.
• The angle at the center of a circle is twice the angle at the circumference subtended by the same arc (central angle theorem).

Diagram (a):

• Angle at the center, marked as angle $A$, subtended by the same arc as the 20° angle at the circumference (let's call it angle $B$).
• According to the central angle theorem: $A = 2 \times B = 2 \times 20^\circ = 40^\circ$ So, angle $a = 40^\circ$.

Diagram (b):

• The central angle is given as 130° (marked as angle $A$).
• The angle at the circumference subtended by the same arc is $b$. $b = \frac{A}{2} = \frac{130^\circ}{2} = 65^\circ$ So, angle $b = 65^\circ$.

Diagram (c):

• The angle at the center is divided into two parts, marked as $a$ and $b$. The angle at the circumference is 30°.
• From the central angle theorem: $a + b = 2 \times 30^\circ = 60^\circ$ Therefore, the sum of angles $a$ and $b$ is 60°. Without additional information, we cannot split $a$ and $b$, but their sum is 60°.

Diagram (d):

• Angle at the center (marked $a$) is subtended by the same arc as the 80° angle at the circumference (marked $c$). $a = 2 \times c = 2 \times 80^\circ = 160^\circ$
• The angle $b$ is at the circumference on the other side, subtended by the same arc as $c$, so: $b = c = 80^\circ$

Summary of Answers:

• (a) $a = 40^\circ$
• (b) $b = 65^\circ$
• (c) $a + b = 60^\circ$
• (d) $a = 160^\circ$, $b = 80^\circ$

Would you like further clarification on any of the steps?

Here are 5 related questions:

1. How is the central angle theorem applied in circle geometry?
2. Can you explain the properties of cyclic quadrilaterals in a circle?
3. How do angles subtended by the same arc behave?
4. What is the relationship between a diameter and an angle in a semicircle?
5. How would you determine an unknown angle if multiple arcs subtend the same angle?

Tip: In circle geometry, always check for symmetry and angles subtended by the same arc to find unknown angles!