**Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.**

**S****olution:**

Let's suppose that we are given a quadrilateral ABCD. Diagonals of quadrilateral ABCD bisect each other at right angles.

In and

**(Given)**

** (Each given equal to )**

**(Given)**

Therefore, by **SAS congruence rule**,

It means that we have AB=CD **(Corresponding parts of congruent triangles are equal)**

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

because and are alternate interior angles.

Quadrilateral is a parallelogram if one pair of opposite sides is equal and parallel. It means that quadrilateral ABCD is a parallelogram. ** (1)**

and ** (2)**

In order to prove that it is a rhombus, we just need to prove that all the sides of parallelogram ABCD are equal.

Now in and

** (Common)**

**(It is given that diagonals bisect each other at )**

**(Given)**

Therefore, **by SAS congruence rule**,

It means that we have AB=AD **(Corresponding parts of congruent triangles are equal) (3)**

From **(1)**, **(2)** and **(3)**, we can say that ABCD is a parallelogram having all the sides equal. It means that ABCD is a rhombus.