2.1 Determine Q 10.2.2 Determine R

Here are the solutions for the angles:

10.2.1 Determine $\hat{Q}$

Step 1: Identify known angles and properties.
We are given that ST is a tangent to the circle at S, and $\angle PST = 42^\circ$.
By the Tangent-Chord Theorem, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
So, $\angle PST = \angle PQS$.

$$\angle PQS = 42^\circ$$

Therefore, $\hat{Q} = \angle PQS$.

$$\boxed{\hat{Q} = 42^\circ}$$

10.2.2 Determine $\hat{