Parallelograms


   
 
Theorem 2
Statement
 
If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram.
 
 
Given:
 
ABCD is a quadrilateral in which diagonals AC and BD
 
intersect at O such that AO=OC and BO=OD.
 
To prove:
 
ABCD is a parallelogram.
 
Proof:
 
 
In triangles AOB and COD,
 
AO = CO (given)
 
BO = OD (given)
 
(vertically opposite angles are equal)
 
D AOB D COD (SAS congruency condition)
 
 
Since these are alternate angles made by the transversal AC intersecting AB and CD.
 
AB||CD
 
Similarly, AD||BC
 
Hence ABCD is a parallelogram.
 
 
     
   
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