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.