Question

abcd is a quadrilateral in which ab=cd and ad=cb show that it is aparallelogram

Answer 1

In quadrilateral ABCD

we have

AB = CD

and

AD = BC

Now join segment

A and C

Now in

ΔABC

and

ΔACD

we have

AB = CD ----- already given

AD = BC ----- already given

and

AC = AC ----- common

Hence

ΔABC ≈ ∆ ACD

∴ <DAC = <BCA - opposite sides are equal

DC and AB

∴

CD || AB - as alternate angles are equal

∴ <DCA = <CAB - opposite sides are equal

AD and BC

∴ AD || BC - as alternate angles are equal

As opposite sides of quadrilateral are parallel,

ABCD is a parallelogram.