Question

Prove that "the opposite angles of a cyclic quadrilateral are supplementary".

Answer 1

Given : A cyclic quadrilateral $ABCD$.
To Prove : $\angle A+\angle C={180}^o$
$\angle B+\angle D={180}^o$
Construction : Let $O$ be the centre of the circle. Join $O$ to $B$ and $D$. Then let the angle subtended by the minor arc and the major arc at the centre be $x^o$ and $y^o$ respectively.
Proof : $x^o=2\angle C$ [Angle at centre theorem] ...(i)
$y^o=2\angle A$ ...(ii)
Adding (i) and (ii), we get
$x^o+y^o=2\angle C+2\angle A$ ...(iii)
But, $x^o+y^o={360}^o$ ....(iv)
From (iii) and (iv), we get
$2\angle C+2\angle A={360}^o$
$\Rightarrow$ $\angle C+\angle A={180}^o$
But we know that angle sum property of quadrilateral
$\angle A+\angle B+\angle C+\angle D={360}^o$
$\angle B+\angle D+{180}^o={360}^o$
$\angle B+\angle D={180}^o$

Hence proved.