Using the theorem "Angle at centre = 2 x Angle at circumference", find x, y, z.

Let's determine the values of $x, y, z$ for each figure using circle theorems. The key theorem provided is: "Angle at centre $= 2 \times$ Angle at circumference".

Figure 1:

Step 1: Identify the angles.
The angle at the center subtending arc BC is $\angle BOC = 100^\circ$.
The angle at the circumference subtending the same arc BC is $\angle BAC = x$.

Step 2: Apply the theorem.

$$\angle BOC = 2 \times \angle BAC$$ $$100^\circ = 2x$$

Step 3: Solve for $x$.

$$$$