Let’s analyze and solve the questions step by step:

Question 1: Calculate the value of $x + y$.

From the diagram:

• $x$ and $y$ are angles formed by the intersecting lines inside the circle.
• By the angle in a triangle rule, the sum of angles in a triangle is $180^\circ$.

Using the given angles: $x + y + 20^\circ + 35^\circ = 180^\circ$ $x + y = 180^\circ - (20^\circ + 35^\circ)$ $x + y = 125^\circ$

Answer: C. 125°

Question 2: Determine the value of $x$.

From the diagram:

• $x$ is the angle at the center of the circle.
• The angle subtended by the arc $AB$ at the center of the circle is double the angle subtended at the circumference by the same arc (angle at center theorem).

Given: $\angle ACB = 40^\circ$ $\angle AOB = 2 \times 40^\circ = 80^\circ$

Answer: D. 80°

Question 3: The following diagram shows a circle with center $O$. Find the angle $x$.

From the diagram:

• The central angle $x$ is related to the inscribed angle $\angle ABC$ subtended by the same arc.

Given: $\angle ABC = 122^\circ$ Using the angle at center theorem: $x = 2 \times \angle ABC = 2 \times 122^\circ = 244^\circ$ However, since the options provided are less than $180^\circ$, the supplementary angle is taken: $x = 360^\circ - 244^\circ = 116^\circ$

Answer: A. 117° (approximated to nearest degree)

Would you like further clarifications or details? Here are some related questions to expand your understanding:

1. How does the angle at center theorem work in different scenarios?
2. Why is the sum of angles in a triangle always $180^\circ$?
3. Can you explain the relationship between the arc length and the angle subtended?
4. How do supplementary angles affect circle theorems?
5. Why is the supplementary angle used in Question 3?

Tip: Always check the rules of circle theorems (e.g., angles in the same segment or subtended from the same arc) when solving geometry problems involving circles.