# 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 ${180}^{\circ}$. ∎

Title | opposing angles in a cyclic quadrilateral^{} are supplementary^{} |
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Canonical name | OpposingAnglesInACyclicQuadrilateralAreSupplementary |

Date of creation | 2013-03-22 17:13:31 |

Last modified on | 2013-03-22 17:13:31 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 8 |

Author | rm50 (10146) |

Entry type | Theorem |

Classification | msc 51M04 |