The problem shown in the image asks to find the angles marked with letters in the given diagrams. In each diagram, $O$ is the center of the circle. Let's analyze the four parts individually to solve for the unknown angles.

(i) Diagram Analysis:

• The angles inside the circle include $30^\circ$, $65^\circ$, and the angle marked as $x$.
• The sum of angles in a triangle is $180^\circ$.

In this diagram, we see two radii forming an isosceles triangle. The base angles are both $65^\circ$, and the vertex angle $x$ can be calculated using: $x = 180^\circ - 2 \times 65^\circ = 180^\circ - 130^\circ = 50^\circ.$

(ii) Diagram Analysis:

• Angles given: $80^\circ$, $40^\circ$, and the unknown angle $y$ at the circumference.
• The angle at the center is double the angle at the circumference subtended by the same arc (Circle Theorem).

Thus, using the circle theorem: $y = \frac{80^\circ}{2} = 40^\circ.$

(iii) Diagram Analysis:

• Given angles: $30^\circ$, $40^\circ$, and $70^\circ$.
• To find angle $z$, we use the fact that the sum of angles in a quadrilateral is $360^\circ$.

The sum of known angles is: $30^\circ + 40^\circ + 70^\circ = 140^\circ.$ Thus, angle $z$ is: $z = 360^\circ - 140^\circ = 220^\circ.$ However, since it is a reflex angle at the center, we subtract $180^\circ$, so: $z = 220^\circ - 180^\circ = 40^\circ.$

(iv) Diagram Analysis:

• Given angles: $68^\circ$, $10^\circ$, and angle $p$ at the center.
• $p$ is the reflex angle at the center subtended by the diameter of the circle, making it: $p = 180^\circ - 68^\circ = 112^\circ.$

If you need more details on any step or have specific questions, feel free to ask!

5 Related Questions:

1. How do you apply the circle theorem to angles subtended by the same arc?
2. Can you explain the properties of angles in an isosceles triangle within a circle?
3. Why does the sum of angles in a quadrilateral equal $360^\circ$?
4. What is the relationship between the angle at the center and the angle at the circumference?
5. How does the reflex angle impact the solution in part (iii)?

Tip: Always check if the triangle formed by the radius is isosceles—this simplifies finding unknown angles.