In Geometry, we have learned different types of polygons, such as triangle, quadrilateral, pentagons and so on. Based on the number of sides, polygons are classified into different types. A polygon with four sides is called a quadrilateral. Some of the examples of quadrilaterals are square, rectangle, rhombus, parallelogram, etc. We are quite familiar with the important properties of these quadrilaterals. Now, we are going to have a discussion about another condition for a quadrilateral to be a parallelogram with the help of an example.
Another Condition for a Quadrilateral to be a Parallelogram – Theorem
We are familiar with some of the important properties of parallelogram, such as:
- The opposite side of a parallelogram are parallel and congruent
- The opposite angles of a parallelogram are equal
- Diagonals bisects each other
- Consecutive angles are supplementary, and so on.
Now, we are going to discuss yet another least required condition for a quadrilateral to be a parallelogram.
Theorem: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.
In a quadrilateral ABCD, the side AB is equal to the side DC.
(i.e) AB = CD
Also, the side AB is parallel to the DC.
(i.e) AB || CD.
Now, draw the diagonal AC.
We know that a diagonal of a parallelogram divides the parallelogram into two congruent triangles.
By using the Side-Angle-Side (SAS rule), we can prove that ∆ ABC ≅ ∆ CDA.
Hence, the side BC is parallel to AD. (i.e) BC || AD.
Therefore, ABCD is a parallelogram.
Example
Now, let us take an example to understand the property of a parallelogram.
Question:
ABCD is a parallelogram. The points P and Q are the mid-points of opposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R. Prove that APCQ, DPBQ, PSQR are parallelograms.
Solution:
Consider a quadrilateral APCQ,
Since, AB is parallel to CD, we can say AP || QC ….(1)
Given that, AP = ½ (AB) and CQ = ½ (CD)
Also, AB = CD
2AP = 2CQ
Now, cancel out “2” on both sides, we get
AP = CQ ….(2)
From (1) and (2), we can conclude that the opposite sides are equal and parallel to each other.
Therefore, a quadrilateral APCQ is a parallelogram.
Similarly, we can prove that quadrilateral DPBQ is a parallelogram, as DQ = PB and DQ || PB.
In a quadrilateral PSQR,
As, SP is a part of DP and QR is a part of QB, SP || QR
Also, SQ || PR.
Therefore, a quadrilateral PSQR is a parallelogram.
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