Let's solve the problem step by step. Here's what we know:

• $O$ is the center of the circle.
• We are asked to calculate the size of angle $x$, which is located at the center of the circle, formed by the radii $ON$ and $OK$.
• $\angle MNK = 80^\circ$ and $\angle MKJ = 126^\circ$ are given.

Applying the Theorems:

1. Theorem 1: The angle at the center is twice the angle at the circumference subtended by the same arc.

   $\angle x$ is subtended by the arc $MK$, and at the circumference, the angle subtended by this arc is $\angle MNK = 80^\circ$.

   According to the theorem, the angle at the center $x$ will be twice the angle at the circumference: $x = 2 \times 80^\circ = 160^\circ$

So, the size of angle $x$ is $160^\circ$.

Would you like more detailed information or clarification on any steps? Here are five related questions:

1. What is the general rule for angles subtended by the same arc at the center and at the circumference of a circle?
2. How can we use cyclic quadrilateral properties in such circle geometry problems?
3. What are the steps to prove the theorem used in this solution?
4. How do inscribed angles behave when subtended by the same arc?
5. Can this method be applied to any circle, regardless of size or specific angles?

Tip: Always keep in mind that angles subtended by the same arc on the circumference are equal, and the center angle will always be double the circumference angle subtended by the same arc!