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The diagram provided shows a circle with points \( A \), \( F \), \( G \), \( D \), and \( B \) on the circumference, and a number of angles labeled in the interior. You are asked to find the values of \( x \), \( y \), and \( z \).

### Step-by-step Explanation:

1. **Identifying the relevant angles:**
   - Angle \( \angle BCF = 43^\circ \).
   - Angle \( \angle AEF = 52^\circ \) (inscribed in the circle).
   - We need to find \( x \) (angle \( \angle AFE \)), \( y \) (angle \( \angle EBG \)), and \( z \) (angle \( \angle EGD \)).

2. **Use of Circle Theorems:**
   - **Inscribed Angle Theorem**: The angle subtended by an arc at the center is twice the angle subtended at the circumference.
   - **Angles in the Same Segment**: Angles subtended by the same arc in the same segment are equal.
   - **Opposite Angles in a Cyclic Quadrilateral**: In any cyclic quadrilateral (a four-sided figure inscribed in a circle), the opposite angles sum up to \( 180^\circ \).

#### Solving for \( x \):
- \( \angle AFE \) (denoted as \( x \)) and \( \angle AEF = 52^\circ \) are in the same segment, so by the **Angles in the Same Segment** theorem, \( x = \angle AEF = 52^\circ \).

#### Solving for \( z \):
- \( \angle EGD \) (denoted as \( z \)) and \( \angle BCF = 43^\circ \) are angles subtended by the same arc in the circle. By the **Angles in the Same Segment** theorem, \( z = 43^\circ \).

#### Solving for \( y \):
- By the **Opposite Angles in a Cyclic Quadrilateral** theorem, \( \angle EBG + \angle AEF = 180^\circ \).
- So, \( y + 52^\circ = 180^\circ \), which gives \( y = 180^\circ - 52^\circ = 128^\circ \).

### Final Answers:
- \( x = 52^\circ \)
- \( y = 128^\circ \)
- \( z = 43^\circ \)

Would you like a more detailed breakdown of any step? 

Here are five related questions for practice:
1. What is the Inscribed Angle Theorem?
2. How do angles in a cyclic quadrilateral behave?
3. How do you find angles subtended by the same arc?
4. What is the sum of angles in a triangle?
5. Can you apply the same rules to a non-cyclic quadrilateral?

**Tip:** Always look for angles subtended by the same arc or chord when working with circle theorems.

Calculate x, y, and z based on the angles in a circle, with some angles and arcs given in the diagram.

Learn how to calculate x, y, and z in a circle using circle geometry theorems. This problem involves the Inscribed Angle Theorem, Angles in the Same Segment, and Cyclic Quadrilateral properties. Ideal for students in Grades 9-12.

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