Question

Prove that the opposite angles of cyclic quadrilateral are supplementary or or sum is 180°.

Answer 1

Answer:

Proof is given above

Step-by-step explanation:

proof is given above.

Answer 2

Given:

A cyclic quadrilateral ABCD.

To Prove: ∠$A+$ ∠$C=180$

                ∠$B+$∠$D=180$°

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 $x$° and $y$°  respectively.

Proof : $x$° $=2$∠$C$      [Angle at centre theorem]    ...(i)

$y$° $=2$∠$C$                                                   ...(ii)

Adding (i) and (ii), we get

          $x$°+$y$° $=2$∠$C+2$∠$A$             ...(iii)

But,     $x$°+$y$°$=360$°                      ....(iv)

From (iii) and (iv), we get

           $2$∠$C$ $+2$∠$A$ $=360$°

⇒            ∠$C$ $+$ ∠$A=180$°

But we know that angle sum property of quadrilateral

∠$A+$ ∠$B+$ ∠$C+$∠$D=360$°

∠$B+$ ∠$D+180$° $=360$°

∠$B$$+$ ∠$D=180$°

Hence proved.