Let's analyze the diagram and the questions one by one.

10.2.1 Give a reason why $\hat{N}_2 = x$.

Looking at the diagram, $\hat{N}_2$ refers to the angle formed at $N$ in the smaller circle, specifically $\angle PNQ$. Since $P, Q, N$ lie on a circle and the arc $PQ$ subtends $\hat{N}_2$, we can deduce that $\hat{N}_2$ is equal to the angle subtended by the same arc at another point on the circumference (in this case, $x$). By the angle subtended by the same arc theorem (angles subtended by the same arc in a circle are equal), we conclude:

• $\hat{N}_2 = x$.

10.2.2 Write down another angle equal in size to $x$. Give a reason.

Another angle equal to $x$ is $\hat{M}_2$ (which refers to $\angle PMS$ in the diagram). The reason is again based on the angle subtended by the same arc theorem. The angle $\hat{M}_2$ is subtended by the same arc $PQ$ as $\hat{N}_2$, so:

• $\hat{M}_2 = x$.

10.2.3 Determine the size of $\hat{R}$ in terms of $x$.

$\hat{R}$ refers to $\angle PRS$. This angle subtends the major arc $PS$ of the larger circle. By the exterior angle theorem for circles, we know that the exterior angle at $R$ is equal to the sum of the interior opposite angles. Therefore, the size of $\hat{R}$ can be determined by:

• $\hat{R} = 2x$, since $\hat{N}_2 = x$ and $\hat{M}_2 = x$.

10.2.4 Prove that $PS = SR$.

The question requires us to prove that line segments $PS$ and $SR$ are equal in length. Observing the diagram, $P$ and $R$ lie on the circumference of a circle, with $S$ as the midpoint. Therefore, by symmetry or the fact that $S$ is the midpoint of the chord subtending the angle $PR$, we can deduce that:

• $PS = SR$.

Would you like further details on any of these explanations or calculations?

Further Related Questions:

1. How can we use the properties of cyclic quadrilaterals to further explore this diagram?
2. What is the significance of the symmetry observed in this diagram?
3. Could you derive a formula for other angles in this diagram using circle theorems?
4. How would the solution change if point $S$ was not the midpoint of $PR$?
5. Can you explain how the angle at the center of the circle relates to the angles on the circumference?

Tip: Always look for symmetry in geometric diagrams, as it often leads to equal lengths or angles that simplify proofs!