The image you uploaded contains a geometry problem involving a circle with points $A$, $B$, $C$, and $D$ on its circumference. The problem is written in both English and another language, and includes a diagram showing a cyclic quadrilateral (a four-sided figure with all vertices on the circumference of a circle).

Here is the relevant information extracted from the image:

Problem Statement:

2. In the figure, O is the center of the circle.

• $A$, $B$, $C$, and $D$ are points on the circle such that $\angle ABC = 40^\circ$.

Questions:

• a) What is the measure of $\angle AOC$?
• b) What is the measure of $\angle ADC$?

Solution:

Given that $\angle ABC = 40^\circ$:

a) Measure of $\angle AOC$:

• $\angle AOC$ is the central angle subtended by the arc $AC$.
• The central angle is twice the inscribed angle subtended by the same arc.
• Therefore, $\angle AOC = 2 \times \angle ABC = 2 \times 40^\circ = 80^\circ$.

b) Measure of $\angle ADC$:

• $\angle ADC$ is the angle subtended by the same arc $AC$ on the opposite side of the circle.
• According to the property of cyclic quadrilaterals, opposite angles sum to $180^\circ$. Since $\angle ABC = 40^\circ$, and $\angle ADC$ is subtended by the same arc, it must be equal to $\angle ABC$.
• Therefore, $\angle ADC = 40^\circ$.

Final Answers:

• a) $\angle AOC = 80^\circ$
• b) $\angle ADC = 40^\circ$

Would you like more details on any part of this solution? Here are five related questions to consider:

1. Why is the central angle twice the inscribed angle in a circle?
2. How does the sum of opposite angles in a cyclic quadrilateral equal $180^\circ$?
3. What is the relationship between the angles subtended by the same arc?
4. How can the inscribed angle theorem be used in different types of problems?
5. What other properties of cyclic quadrilaterals are useful in problem-solving?

Tip: Always remember that the central angle theorem is a powerful tool in circle geometry, as it directly relates the central angle to the inscribed angle subtended by the same arc.