$ABCD$ is a cyclic quadrilateral inscribed in a circle. If $\angle A$ measures $75^{\circ}$, what is the measure of $\angle C$?
If $\angle B$ measures $110^{\circ}$, what is the measure of $\angle D$?

In a cyclic quadrilateral, opposite angles sum to $180^\circ$.

$$\angle A + \angle C = 180^\circ \Rightarrow \angle C = 180^\circ - 75^\circ = 105^\circ.$$

$$\angle B + \angle D = 180^\circ \Rightarrow \angle D = 180^\circ - 110^\circ = 70^\circ.$$

Caution
Students should remember that in a cyclic quadrilateral only the opposite angles are supplementary (sum to $180^{\circ}$), not the adjacent angles.