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In this problem, you're asked to calculate the angles indicated on the diagram of a circle with several lines and angles.

### Key observations:

- The given angles are inside or around a cyclic quadrilateral, meaning some angles can be computed using circle theorems, particularly:
  1. **Inscribed angle theorem**: The angle subtended by an arc at the circumference is half the angle subtended at the center by the same arc.
  2. **Opposite angles in a cyclic quadrilateral** sum to 180°.

- **Angle MCE = 36°** (already given in the diagram).
- **Angle BME = 110°** (given in the diagram as well).

Now let's solve for the unknown angles:

1. **Angle A (at point A)**:
   Since this is an inscribed angle (subtended by the arc BC), we can apply the inscribed angle theorem:
   \[
   \text{Angle A} = \frac{1}{2} \times \text{Angle BMC}
   \]
   \[
   \text{Angle A} = \frac{1}{2} \times 110° = 55°
   \]

2. **Angle B (at point B)**:
   Since angles subtended by the same arc (in this case, arc DC) are equal, angle B is equal to angle MCE:
   \[
   \text{Angle B} = 36°
   \]

3. **Angle D (at point D)**:
   Using the inscribed angle theorem again, angle D subtends the arc BC. Therefore:
   \[
   \text{Angle D} = \frac{1}{2} \times \text{Angle BMC}
   \]
   Since angle BMC is 110°:
   \[
   \text{Angle D} = \frac{1}{2} \times 110° = 55°
   \]

4. **Angle E (at point E)**:
   The angle at point E is vertically opposite angle MCE, so:
   \[
   \text{Angle E} = 36°
   \]

### Final answers:
- **Hoek A = 55°**
- **Hoek B = 36°**
- **Hoek D = 55°**
- **Hoek E = 36°**

Would you like more detailed explanations for any of these angles? Here are 5 related questions for further exploration:

1. What is the relationship between opposite angles in a cyclic quadrilateral?
2. How does the inscribed angle theorem apply to different arcs in a circle?
3. How can we calculate angles in a triangle inscribed in a circle?
4. Why are angles subtended by the same arc equal in a circle?
5. How can we use the central angle to compute related inscribed angles?

**Tip**: When dealing with angles in a circle, always look for arcs that subtend those angles. This will guide you in applying the right theorems!

Bereken de grootte van de aangeduide hoeken (Hoek A, Hoek B, Hoek D, Hoek E) in de cirkel.

Learn how to calculate the angles in a circle using the inscribed angle theorem and properties of cyclic quadrilaterals. This problem involves finding the angles Hoek A, Hoek B, Hoek D, and Hoek E using key geometric concepts and theorems. Suitable for students in Grades 9-10.

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