Let's analyze the problem from the image you uploaded.

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.