Question

Explain: If a quadrilateral is cyclic then the opposite angles are supplementary

Answer 1

The theorem states that if a quadrilateral is cyclic its opposite angles and supplementary. This means following:
i) A quadrilateral is cyclic - This means that all the 4 vertices of a quadrilateral lie on circumference of a circle or there is circle circumscribing a quadrilateral.
(ii) In such a quadrilateral, the sum of opposite angles is 180 degrees.
Now, how is the sum of opposite angles 180 degrees ? Proof as below:

ABCD is a cyclic quadrilateral. Now Join AC and also join A to O and C to O.
AC is a chord and therefore, $\angle AOC$ = 2 $\times$ $\angle ABC$ (angle at the centre is 2 $\times$ angle in segment) ..(i)
Similarly Reflex $\angle AOC$ = 2 $\times$ $\angle ADC$ (ii)
Adding (i) and (ii), we get,
2 $(\angle ABC+\angle ADC)$ = $\angle AOC$ + Relex $\angle AOC$ = 360
$\Rightarrow$ $\angle ABC+\angle ADC$ = 180 degrees.
So, opposite angles are complimentary.