In die diagram is O die middelpunt van die sirkel - NSC Mathematics - Question 9 - 2023 - Paper 2

We note that the angle at the circumference $\angle O_1$ subtended by the arc AC is equal to twice the angle at the center, according to the inscribed angle theorem. Thus, we have:

$\angle O_1 = 2B$.

Similarly, the angle $\angle O_2$ subtended by the arc BD is also equal to twice the angle at the center. Therefore:

$\angle O_2 = 2D$.

Adding the angles $\angle O_1$ and $\angle O_2$, we can express the revolution at the center of the circle:

$\angle O_1 + \angle O_2 = 360^\circ$.

From the previous equations, substituting the expressions we derived:

$2B + 2D = 360^\circ$. Dividing everything by 2 leads us to: $B + D = 180^\circ.$ This confirms that the opposite angles in a cyclic quadrilateral are supplementary.