Question

Prove that a parallelogram is not cyclic which is not a rectangle??

Answer 1

Please see below.

Explanation:

Let the parallelogram be `ABCD`.

As opposite angles of a parallelogram are equal, we have

`m/_A=m/_C` and `m/_B=m/_D`

further adjacent angles are supplementary and hence add up to `180^@`.

In case parallelogram is cyclic, as opposite angles of a cyclic quadrilateral are supplementary i.e. they add up to `180^@`,

we have `m/_A+m/_C=180^@`

but as it is also a parallelogram, they are equal too and then each must be `90^@`. Hence parallelogram if cyclic, is a rectangle too.

However, if it is not a rectangle, opposite angles will not add up to `180^@` and parallelogram will not be cyclic.

`Q.E.D.`