https://file.aiquickdraw.com/mathbot/file/1729181049729qtiqrmh6.jpg

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!

10.2.1 Give a reason why ∠N2 = x. 10.2.2 Write down another angle equal in size to x. Give a reason. 10.2.3 Determine the size of ∠R in terms of x. 10.2.4 Prove that PS = SR.

Explore a complex geometry problem involving circle theorems. Learn how angles subtended by the same arc and the exterior angle theorem can be applied to solve questions about equal angles and symmetry. Includes detailed proofs and explanations for angles and line segments within circles.

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