Question

Prove that opposite angles of a cyclic quadrilateral are supplementary:

Answer 1

To prove that ABCD is a cyclic quadrilateral :

$\angle A+\angle C=180°and\angle B+\angle D=180°$

Construction:

Let the centre of the circle be O. Join O to B and D. Let x and y be the angle subtended by the minor arc and the major arc respectively.

Now,

$$\begin{gathered} \Rightarrow x°=2\angle C \\ \Rightarrow y°=2\angle A \end{gathered}$$

Adding $x°+y°=2\angle C+2\angle A$

As we get $x°+y°=360°\left(Completeangle\right)$

Therefore, $\angle C+\angle A=180°$

From the property of quadrilateral, we know that,

$$\begin{gathered} \Rightarrow \angle A+\angle B+\angle C+\angle D=360° \\ \Rightarrow \angle B+\angle D+180°=360° \\ \Rightarrow \angle B+\angle D=180° \end{gathered}$$

Hence, proved