A Quadrilateral is a Parallelogram if its Diagonals Bisect Each Other

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.

Also Read:

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?
A Quadrilateral is a Parallelogram 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.

Leave a Reply