Parallelograms


   
 
Theorem 1
Statement
 
The diagonals of a parallelogram bisect each other.
 
 
Given:
 
ABCD is a parallelogram in which diagonals AC and BD
 
intersect each other at O.
 
To prove:
 
The diagonals AC and BD bisect each other i.e, AO=OC and BO=DO.
 
Proof:
 
 
AB||CD (by definition of parallelogram)
 
AC is a transversal.
 
 
 
(alternate angles are equal in a parallelogram)
 
Also AB=DC (opposite sides are equal in a parallelogram)
 
Now in D AOB and D COD,
 
AB=DC (opposite sides of parallelogram are equal)
 
(proved by (i))
 
 
 
D AOBD COD (AAS congruency condition)
 
AO=OC and OB=OD (corresponding parts of congruent triangles are congruent)
 
 
i.e., the diagonals of a parallelogram bisect each other.
 
We have proved one of the properties of a parallelogram by logical deductions.
 
 
     
   
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