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### a) Finding the size of angle \( x \):

We are given a diagram with a circle and a tangent, where:
- One of the angles in the triangle is \( 29^\circ \),
- Another angle is \( 53^\circ \),
- The third angle is \( x \), and it is an exterior angle with respect to the triangle.

The theorem that will help solve this is the **"alternate segment theorem."** This theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment (on the opposite side of the chord). 

Thus, angle \( x \) is equal to the angle \( 29^\circ \) (as per the alternate segment theorem).

So, the size of angle \( x = 29^\circ \).

### b) Choosing the theorem:
As mentioned above, the correct theorem that allows us to find \( x \) is the **alternate segment theorem**.

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

Here are five related questions to deepen understanding:
1. What is the alternate segment theorem, and how is it applied in different cases?
2. Can the angle sum property of a triangle be used to find angle \( x \) if the triangle were complete?
3. How do the properties of tangents relate to angles formed inside a circle?
4. What is a cyclic quadrilateral, and how do its opposite angles relate to circle theorems?
5. How does the exterior angle theorem of a triangle compare with the alternate segment theorem?

### Tip:
When dealing with circles and tangents, remember that the angle between a tangent and a radius at the point of contact is always \( 90^\circ \). This can often help solve complex circle-related problems.

The diagram shows a circle with a tangent. a) What is the size of angle x? b) Choose the theorem below which allows you to find x.

Learn how to find the angle x in a circle using the alternate segment theorem. This problem involves key concepts such as circle theorems, angles in triangles, and tangents. Suitable for students in Grades 8-10.

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