# 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 $180^{\circ}$.

###### Proof.

Let $ABCD$ be a quadrilateral inscribed in a circle

Note that $\angle BAD$ subtends arc $BCD$ and $\angle BCD$ subtends arc $BAD$. Now, since a circumferential angle is half the corresponding central angle, we see that $\angle BAD+\angle BCD$ 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

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

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

Title opposing angles in a cyclic quadrilateral are supplementary OpposingAnglesInACyclicQuadrilateralAreSupplementary 2013-03-22 17:13:31 2013-03-22 17:13:31 rm50 (10146) rm50 (10146) 8 rm50 (10146) Theorem msc 51M04