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.