Rhombus: A parallelogram having all sides equal is called a rhombus.

Thus, a parallelogram ABCD is a rhombus if adjacent sides AB = AD.

In other words, a quadrilateral ABCD is a rhombus if   and

       AB = BC = CD = DA.

Kite: A quadrilateral ABCD is a kite, if AB= AD, BC = CD but AD BC and AB CD.

It follows from the above definitions that:

(i) A square, rectangle and rhombus are parallelograms.

(ii) A parallelogram is a trapezium but a trapezium but a trapezium is not a parallelogram.

(iii) A rectangle or a rhombus is not necessarily a square.

(iv) A kite is not a parallelogram.

Given: A rhombus ABCD, such that AB = BC

To Prove: AB = BC = CD= DA

Proof: ABCD is a rhombus.

As every rhombus is a Parallelogram ABCD is a ||gm.

 AB = CD and AD = BC    [Opposite Parallelogram are equal]

But AB = BC    [Given]

AB = BC = CD= DA

Hence Proved

  AB = AD       [ Sides of a Rhombus ABCD]

  BC = DC       [ Sides of a Rhombus ABCD]

 AC = AC       [ Common ]

Therfore ABC  ADC     [SSS Criteria of Congruence]

       [ By CPCT ]    ............(3)

and      [ By CPCT ]

Hence diagonal AC bisects A and C