opposing angles in a cyclic quadrilateral are supplementary

Theorem 1.

[Euclid, Book III, Prop. 22] If a quadrilateral is inscribed in a circle, then opposite angles of the quadrilateral sum to ${\mathrm{180}}^{\mathrm{\circ }}$.

Proof.

Let $ABCD$ be a quadrilateral inscribed in a circle

Note that $\mathrm{\angle }BAD$ subtends arc $BCD$ and $\mathrm{\angle }BCD$ subtends arc $BAD$. Now, since a circumferential angle is half the corresponding central angle, we see that $\mathrm{\angle }BAD+\mathrm{\angle }BCD$ is one half of the sum of the two angles $BOD$ at $O$. But the sum of these two angles is ${360}^{\circ }$, so that

$$\mathrm{\angle }BAD+\mathrm{\angle }BCD={180}^{\circ }$$

Similarly, the sum of the other two opposing angles is also ${180}^{\circ }$.∎