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Let's break down the problem step by step to find the size of angle \( x \).

### Given Information:
- \( A \) and \( C \) are points on a circle with center \( O \).
- \( AB \) and \( CB \) are tangents to the circle.
- The angle \( \angle ABC = 52^\circ \).
- We are asked to work out the size of angle \( x \), which is \( \angle AOC \).

### Key Theorems and Facts:
1. **Tangent-Radius Theorem**: The angle between a tangent and the radius at the point of contact is \( 90^\circ \). So, \( \angle OAB = 90^\circ \) and \( \angle OCB = 90^\circ \).
   
2. **Tangent-Secant Angle Theorem**: The exterior angle formed by two tangents to a circle from a point outside the circle is equal to the angle subtended by the two radii at the center of the circle. This means that:
   \[
   \angle AOB = \angle ABC = 52^\circ.
   \]

3. **Isosceles Triangle**: Since \( OA = OC \) (both are radii of the circle), triangle \( AOC \) is isosceles. Therefore, the two base angles \( \angle OAC \) and \( \angle OCA \) are equal.

### Solution Steps:
1. **Work out angle \( AOC \)**:
   The angle at the center of the circle \( \angle AOC \) is the angle we're trying to find, which can be determined using the fact that the total angle around point \( O \) in the triangle is \( 360^\circ \).

   Since the two base angles \( \angle OAB \) and \( \angle OCB \) are each \( 90^\circ \), we know:
   \[
   \angle AOC = 180^\circ - \angle ABC.
   \]
   Given \( \angle ABC = 52^\circ \), we find:
   \[
   \angle AOC = 180^\circ - 52^\circ = 128^\circ.
   \]

Thus, the size of angle \( x = \angle AOC = 128^\circ \).

### Final Answer:
The size of angle \( x = 128^\circ \).

Would you like any further details or clarifications on the steps? Here are 5 related questions for further practice:

1. What is the measure of \( \angle OAC \) if triangle \( AOC \) is isosceles?
2. How do we determine the angle subtended by a chord at the center of a circle?
3. Why is \( \angle OAB = 90^\circ \) in the diagram provided?
4. What is the sum of angles in a triangle, and how does it help in solving this problem?
5. Can you explain why the tangents from a point outside the circle are equal in length?

**Tip**: In circle geometry, always remember that angles subtended by the same arc at the center are twice the angles subtended on the circumference.

Work out the size of angle x, given that A and C are points on a circle with center O. AB and CB are tangents to the circle, and angle ABC = 52°. Provide a reason for each step.

This problem involves finding the size of angle x in a geometric figure that includes tangents and circle theorems. Angle ABC is given as 52°, and the solution explores key theorems like the Tangent-Radius Theorem and properties of isosceles triangles. Suitable for students in Grades 9-11.

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