**Diagonal AC of a parallelogram ABCD bisects (See figure). Show that **

**(i)** it bisects also,

**(ii)** ABCD is a rhombus.

**Solution (i)**

It is given that ABCD is a paralleogram and **(1)**

and ** (Opposite sides of parallelogram are parallel)**

Therefore, ** (Alternate Interior Angles) (2)**

And, **(Alternate Interior Angles) (3)**

From **(1), (2)** and **(3)** we can say that diagonal AC also bisects .

**Solution (ii)**

In and

** (Proved above in solution (i))**

AC=AC ** (Common)**

**(Given)**

Therefore,** by ASA congruence rule**,

**(Corresponding parts of congruent triangles are equal) (4)**

We already have AB=CD and BC=AD because ABCD is a parallelogram. ** (5)**

From **(4)** and **(5)**, we can say that ABCD is a parallelogram having all the sides equal. Therefore, ABCD is a rhombus.