https://file.aiquickdraw.com/mathbot/file/17273739217889iwkzaup.jpg

Let's analyze the image and solve the problem based on the information given.

### Problem Breakdown:
The problem asks to find the value of \( x \). The figure shows a cyclic quadrilateral with certain angles marked.

1. **Key Information:**
   - \(\angle AOB = 135^\circ\)
   - \(\angle AOE = 20^\circ\)
   - \(\angle BOC = 4x\)
   - \(\angle EOC = 2x\)

### Approach:

1. **Use of Angle Sum Property in a Circle:**
   For a quadrilateral inscribed in a circle, opposite angles sum to \(180^\circ\).

2. **Angle Relationship at Point \(O\):**
   Since \(AOBE\) forms a quadrilateral, and \(O\) is the center, we can use the properties of angles subtended by arcs and solve for \(x\).

### Step-by-step Solution:

1. **Angle at Point \(O\):**
   \[
   \text{Sum of all angles at point O} = 360^\circ
   \]
   Thus,
   \[
   \angle AOB + \angle AOE + \angle BOC + \angle EOC = 360^\circ
   \]
   Plugging in the given values:
   \[
   135^\circ + 20^\circ + 4x + 2x = 360^\circ
   \]
   Simplifying:
   \[
   155^\circ + 6x = 360^\circ
   \]
   \[
   6x = 360^\circ - 155^\circ
   \]
   \[
   6x = 205^\circ
   \]
   \[
   x = \frac{205^\circ}{6}
   \]
   \[
   x = 34.17^\circ
   \]

### Answer:
The value of \(x\) is approximately **34.17°**.

Would you like more details or have further questions?

Here are some related questions to expand on the topic:

1. What are the properties of a cyclic quadrilateral?
2. How can you calculate the angles subtended by an arc in a circle?
3. What is the significance of opposite angles in a cyclic quadrilateral?
4. Can this type of problem appear with different configurations of angles?
5. How does the sum of angles around a point work in geometrical figures?

**Tip:** Always double-check your angle calculations, especially when dealing with multiple unknowns in a geometric figure.

Find the value of x, given angles at point O in a cyclic quadrilateral.

Learn how to calculate the unknown angle x in a cyclic quadrilateral, using the sum of angles at a point and cyclic quadrilateral theorems. This geometry problem involves angles such as 135°, 20°, and unknowns like 4x and 2x.

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