Identify the angle at the centre and the angle at the circumference subtended by the same arc.

Here are the solutions to the geometry problems.

2.1 Calculate the size of $\hat{S}$.

Step 1: Identify the angle at the centre and the angle at the circumference subtended by the same arc.
The angle at the centre subtended by arc AB is $\text{A}\hat{\text{O}}\text{B} = 140^\circ$.
The angle at the circumference subtended by the same arc AB is $\hat{S}$.

Step 2: Apply the theorem that the angle at the centre is twice the angle at the circumference.

$$\hat{S} = \frac{1}{2} \times \text{A}\hat{\text{O}}\text{B}$$ $$$$