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To calculate the angle \(x\), we will use key circle theorems:

1. **Theorem Used**: The *angle at the center* of a circle is twice the *angle at the circumference* when subtended by the same arc.

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### Step-by-Step Solution:

- **Step 1**: Identify which angles are subtended by the same arc.

  - Angle \( \angle NKJ = 128^\circ \) (at the circumference) and angle \(x\) (at the center \(O\)) are subtended by arc \(NJ\).

- **Step 2**: Apply the theorem:

  The angle at the center is twice the angle at the circumference.

\[
x = 2 \times \angle NKJ
\]

Substitute the value of \( \angle NKJ \):

\[
x = 2 \times 128^\circ
\]

\[
x = 256^\circ
\]

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### Final Answer:

The size of angle \(x\) is **256°**.

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Do you want further clarification or additional explanations? Let me know!  

Here are 5 related questions to explore:
1. How do you prove that the angle at the center is twice the angle at the circumference?
2. What is the sum of interior angles in a quadrilateral inscribed in a circle?
3. Can you find other relationships between angles in the circle diagram?
4. What happens if the quadrilateral is cyclic? What properties arise?
5. How can the circle theorems be applied to find missing angles in other geometric figures?

**Tip**: Remember that the angle at the center is always twice the angle at the circumference when subtended by the same arc.

O is the centre of this circle. Calculate the size of angle x. Show all of your working and state any theorems that you use.

Solve for angle x in a circle where O is the center. Learn how to use the theorem that the angle at the center is twice the angle at the circumference subtended by the same arc. A step-by-step solution is provided for students in Grades 9-10.

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