Use the Tangent-Chord Theorem.

Here are the proofs for the given statements:

3.1 Prove that $\hat{S} = x$

Step 1: Use the Tangent-Chord Theorem.
The angle between the tangent $VP$ and the chord $PR$ is $\hat{P}_1$.

$$\hat{P}_1 = \angle RQP \quad (\text{Tangent-Chord Theorem})$$

Given $\hat{P}_1 = x$, so $\angle RQP = x$.

Step 2: Use properties of parallel lines.
We are given that $VTS \parallel RQ$.
Consider $QS$ as a transversal intersecting the parallel lines $RQ$ and $VS$.

$$$$