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To determine the size of angle \( \angle E \) in the diagram, let's analyze the geometry of the situation, especially the cyclic quadrilateral \( ABCE \) and any relationships between the angles provided.

### Key observations:
1. \( \angle AMC = 110^\circ \) is given.
2. \( \angle ACB = 36^\circ \).
3. \( E \) lies on the line segment between \( D \) and \( B \), and since \( ABCE \) is inscribed in a circle, we can apply properties of cyclic quadrilaterals.

### Step-by-Step Calculation:
- **Angles in a cyclic quadrilateral**: Opposite angles in a cyclic quadrilateral add up to \( 180^\circ \). Since \( ABCE \) is a cyclic quadrilateral:
  \[
  \angle AEB + \angle ACB = 180^\circ
  \]
  Substituting \( \angle ACB = 36^\circ \):
  \[
  \angle AEB = 180^\circ - 36^\circ = 144^\circ
  \]

Thus, the size of \( \angle E \) (or more specifically \( \angle AEB \)) is **144°**.

Would you like any further clarifications or details on the process?

### Related Questions:
1. How can we use the inscribed angle theorem to calculate other angles in the diagram?
2. What is the relationship between the central and inscribed angles in this circle?
3. Can the property of the cyclic quadrilateral help in calculating \( \angle D \)?
4. What role does \( \angle AMC = 110^\circ \) play in this circle's geometry?
5. How does the location of point \( M \) affect the rest of the angles?

### Tip:
When working with cyclic quadrilaterals, remember that opposite angles always sum to \( 180^\circ \). This can simplify many calculations!

Learn how to calculate angle E in a cyclic quadrilateral inscribed in a circle. Using key geometric principles such as the cyclic quadrilateral theorem, we determine that the angle measures 144°. Suitable for students in Grades 9-11.

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