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

Here we will discuss about a quadrilateral is a parallelogram if its diagonals bisect each other.

Given: PQRS is a quadrilateral whose diagonals PR and QS bisect each other at O, i.e., OP = OR and OQ = OS.

To prove: PQRS is a parallelogram.

Proof: In ∆OPQ and ∆ORS,

OP = OR, OQ = OS (Given);

∠POQ = ∠ROS (Opposite angles).

Therefore, ∆OPQ ≅ ∆ORS (by SAS criterion of congruency)

Therefore, ∠OPQ = ∠ORS (CPCTC).

So, PQ ∥ SR (From equal alternate angles).

Similarly, from ∆OQR and ∆OSP we get PS ∥ QR.

Therefore, PQRS is a parallelogram.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Successor and Predecessor | Successor of a Whole Number | Predecessor

May 24, 24 06:42 PM

The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

2. ### Counting Natural Numbers | Definition of Natural Numbers | Counting

May 24, 24 06:23 PM

Natural numbers are all the numbers from 1 onwards, i.e., 1, 2, 3, 4, 5, …... and are used for counting. We know since our childhood we are using numbers 1, 2, 3, 4, 5, 6, ………..

3. ### Whole Numbers | Definition of Whole Numbers | Smallest Whole Number

May 24, 24 06:22 PM

The whole numbers are the counting numbers including 0. We have seen that the numbers 1, 2, 3, 4, 5, 6……. etc. are natural numbers. These natural numbers along with the number zero

4. ### Math Questions Answers | Solved Math Questions and Answers | Free Math

May 24, 24 05:37 PM

In math questions answers each questions are solved with explanation. The questions are based from different topics. Care has been taken to solve the questions in such a way that students