Loading....
Coupon Accepted Successfully!

 

Conditions for Quadrilateral to be Parallelogram

Theorem 5
A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.
Given: In Quadrilateral ABCD, AB = DC and AB ïï DC.
To prove: Quadrilateral ABCD is a parallelogram.
Construction: Join B to D.

 


Proof :
In
ΔABD and ΔCDB,
AB = DC   ...... (given)
BD = BD  ...... (common)

ABD = BDC ...... (alternate Interior angles of ABïïDC)
..
ΔABD ΔCDB (SAS Congruence Criteria)
..
ADB = CBD
But
ADB and CBD are alternate interior angles
of AD and BC with transversal BD.
...        AD
ïï BC
Thus, AB
ïï DC and AD ïï BC
... Quadrilateral ABCD is a parallelogram.

To summarise: A quadrilateral is a parallelogram if:
(i)  both pair of opposite sides are equal.
(ii)  both pair of opposite angles are equal.
(iii) diagonals bisect each other.
(iv) a pair of opposite sides is equal as well as parallel.


 

Example :

In a parallelogram ABCD, X and Y are the mid-points of sides AB and DC respectively. Show that DXBY is a parallelogram.

Solution

Given: In a parallelogram ABCD, X is the mid-point of AB (AX = XB) and Y is the mid-point of DC (DY = YC).

To prove: DXBY is a parallelogram.

 

Proof:
AB = DC......(opposite sides of a
ïïgm)
 AB = DC
XB = DY .....(i)
Also      AB
ïï DC..... (opposite sides of ïïgm)
...         XB ïï DY...... (ii)
Now in quadrilateral DXBY, we have XB = DY and XB
ïï DY
...DXBY is a parallelogram.

 





Test Your Skills Now!
Take a Quiz now
Reviewer Name