opposing angles in a cyclic quadrilateral are supplementary

Theorem 1.

[Euclid, Book III, Prop. 22] If a quadrilateralMathworldPlanetmath is inscribedMathworldPlanetmath in a circle, then opposite angles of the quadrilateral sum to 180.


Let ABCD be a quadrilateral inscribed in a circle


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


Similarly, the sum of the other two opposing angles is also 180. ∎

Title opposing angles in a cyclic quadrilateralMathworldPlanetmath are supplementaryPlanetmathPlanetmath
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