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# 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 $180^{{\circ}}$. ∎

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## Mathematics Subject Classification

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