ABCD CLASSES
Another Condition for a Quadrilateral to be a Parallelogram
Another Condition for a Quadrilateral to be a Parallelogram
Theorem 9: A quadrilateral is a parallelogram if a pair of the opposite side is equal and parallel.
Given: A quadrilateral ABCD in which AB = CD and, AB ∥ CD.
To prove: Quadrilateral ABCD is a parallelogram.
Construction: Join BD.
Proof: Now, in ∆ BAD and ∆ DCB, we have
AB = CD (Given)
Since AB ∥ CD and transversal BD intersects at B and D, so alternate interior angles are equal.
⟹ ∠ CDB = ∠ ABD
BD = DB (Common)
Therefore, ∆ BAD ≅ ∆ DCB (By SAS-criterion of congruence)
By using corresponding parts of congruent triangles
⟹ ∠ ADB = ∠ CBD
Now, line BD intersects AB and DC at B and D respectively, such that ∠ ADB = ∠ CBD
That is, alternate interior angles are equal.
∴ AD ∥ BC.
Thus, AB ∥ CD and AD ∥ BC.
Hence, quadrilateral ABCD is a parallelogram.
Example: In the figure, ABCD is a parallelogram and X, Y are the mid- points of sides AB and DC respectively. Show that quadrilateral DXBY is a parallelogram.
Given: ABCD is a parallelogram in which X and Y are the mid-points of AB and DC respectively.
To prove: Quadrilateral DXBY is a parallelogram.
Construction: Join DX and BX.
Proof: Since X and Y are the mid-points of DC and AB respectively.
∴ YB =
But, AB = DC [∵ ABCD is a parallelogram]
⟹ 
AB = DC
⟹ YB = DX. [From(I)] ............. (II)
Also, AB ∥ DC [∵ ABCD is a parallelogram]
∴ YB ∥ DX ............. (III)
Since a quadrilateral is a parallelogram if a pair of the opposite side is equal and parallel.
From (II) & (III), we get Quadrilateral DXBY is a parallelogram.
