We are asked to calculate the size of angle $x$ in the given diagram, with the center of the circle being $O$. The provided information includes the angle $\angle DCE = 136^\circ$.

Explanation:

1. Angles in a Cyclic Quadrilateral: The points $A$, $B$, $C$, and $D$ appear to form a cyclic quadrilateral (all points are on the circumference of the circle). In any cyclic quadrilateral, the opposite angles sum to $180^\circ$.

   So: $\angle ABC + \angle ADC = 180^\circ$ From the diagram, $\angle ADC = 136^\circ$, so: $\angle ABC + 136^\circ = 180^\circ$ Solving for $\angle ABC$: $\angle ABC = 180^\circ - 136^\circ = 44^\circ$

2. Angle at the Center: Additionally, the angle subtended at the center by the same arc is twice the angle subtended at the circumference (this is known as the angle at the center theorem). Therefore, $\angle AOC$ is twice $\angle ABC$: $\angle AOC = 2 \times 44^\circ = 88^\circ$

Thus, the size of angle $x$ is $44^\circ$.

Would you like more details on this solution, or have any specific questions?

Related Questions:

1. What is the sum of the interior angles of a cyclic quadrilateral?
2. How does the angle subtended by the same arc at the center and at the circumference compare?
3. How can the opposite angles in a cyclic quadrilateral be used to solve problems?
4. What other properties of cyclic quadrilaterals can be helpful for solving similar problems?
5. Why is the center of the circle important in angle calculations?

Tip: Remember that angles in a semicircle always measure $90^\circ$. This property can often help when dealing with problems involving circles.