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Converse of Theorem 4

"If the diagonals of a quadrilateral bisect each other, then it is a parallelogram."

 

Given: A quadrilateral ABCD in which AO = OC and BO = OD.

To prove: Quadrilateral ABCD is a Parallelogram. 
 

Proof :
In
ΔAOB and ΔCOD,
AO = OC...... (given)
DO = OB..... (given)
and
AOB = DOC...... (Vertically opposite angles)
...
ΔABO ΔCOD...... (SAS Criterian)

...
ABO = ODC
...ABD = BDC

Since these are the alternate interior angles made by the transversal BD intersecting AB and DC.
AB
ïï DC
Similarly AD
ïï BC
Quadrilateral ABCD is a parallelogram.

 

Example :
In a parallelogram ABCD, M and N are the points on diagonal DB such that DM = NB. Prove that ANCM is a parallelogram.
Solution :

Given: ïïgm ABCD, in which DM = NB
To Prove: ANCM is a parallelogram.
Construction: Join AC.

 


Proof :
As the diagonals of the
ïïgm bisect each other
AO = OC and DO = OB
And DM = BN
bisect each other
AO = OC and DO = OB
And DM = BN
Diagonals bisect each other
AO = OC and DO = OB
And DM = BN

... DO - DM = OB - BN
... MO = ON
Now in quadrilateral ANCM, we have
AO = OC and MO = ON
... diagonals of this quadrilateral bisect each other.
... ANCM is a parallelogram.





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