In the previous article, we have proved that the Diagonals of a Parallelogram Bisect Each Other. The definition of a parallelogram is that the opposite sides are non-intersecting or parallel. It is easy to show that the opposite sides are parallel, thus we can use the definition to prove the theorem and conclude that the figure is a parallelogram.

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## A Quadrilateral is a Parallelogram if its Diagonals Bisect Each Other

**Theorem:**

Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other?

Since it is given that the condition “if and only if”, there are two things to prove.

1. Given,

ABCD is a parallelogram

To prove:

AD and BC intersect at E.

AE = EC, BE = ED

Given

ABCD is a parallelogram

AB || CD [Definition of parallelogram]

∠BAE ≅ ∠DCE [Alternate interior angles]

AB ≅ CD [opposite sides in a parallelogram]

ΔABE ≅ ΔDCE [Angle Side Angle]

AE ≅ EC

BE ≅ ED

And the converse:

Given: AE = EC, BE = ED

To Prove:

ABCD is a parallelogram

AE ≅ EC

BE ≅ ED

∠AEB ≅ ∠CED [vertical angles]

∠AED ≅ ∠CEB [vertical angles]

ΔABC ≅ ΔCDA

ΔABC ≅ ΔCDA

AB ≅ CD

AD ≅ BC

Hence proved

Therefore, a quadrilateral is a parallelogram if and only if the diagonals bisect each other.

### FAQs on Diagonals of a Quadrilateral Bisect Each Other to form a Parallelogram

**1. What is a quadrilateral with diagonals that bisect each other?**

A quadrilateral whose diagonals bisect each other at right angles is a rhombus.

**2. Is it true that if the diagonals of a quadrilateral bisect each other then the quadrilateral is a square?**

A quadrilateral that has diagonals that bisect and are perpendicular must be a square. A kite with congruent diagonals is a square.

**3. Which Quadrilaterals diagonals do not bisect each other?**

Trapezium and Parallelogram are two quadrilateral whose diagonals do not bisect each other at right angles.