O is the centre of circle ABC. AC is produced to D such that CD = CB. If O1 = 128 degrees, prove that D = 32 degrees.

Here are the solutions to the geometry problems:

Problem (i):

• Given: O is the centre of circle ABC. AC is produced to D such that CD = CB. $\hat{O}_1 = 128^\circ$.
• To prove: $\hat{D} = 32^\circ$.

Step 1: Find the angle at the circumference subtended by arc AB.

$$\angle ACB = \frac{1}{2} \times \hat{O}_1$$ $$\angle ACB = \frac{1}{2} \times 128^\circ$$ $$\angle ACB = 64^\circ$$

Reason: Angle at the circumference is half the angle at the centre subtended