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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.

Find the angles marked with letters in the given circle diagrams. In each case, O is the center of the circle.

Solve for unknown angles in circle diagrams using circle theorems and angle properties. This problem includes key concepts like the sum of angles in a triangle and the relationship between angles at the center and circumference, suitable for students in Grades 8-10.

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