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Theorem 4 – In a Parallelogram, the Diagonals Bisect Each Other

Given: In a parallelogram ABCD, the diagonals AC and BD intersect each other at O

To prove: AO = OC and BO = OD

 


Proof: In ΔAOB and ΔCOD, we have
 
BAO = OCD......(Alternate interior angles of AB ïï DC)
 
ABO = CDO...... (alternate interior angles)
  AB = DC ...... (opposite sides of
ïï sides)
...
ΔAOB ΔCOD (ASA Congruence Criteria)
 ...  AO = OC and DO = OB..... (CPCT)

 

Example

The diagonals of a parallelogram ABCD intersect at O. A line through O intersects AB at X and DC at Y. Show that OX = OY.

Solution

ABCD is a parallelogram
... AB
ïï DC
Also BD is a transversal of AB
ïï DC
...
1 = 2[ Alternate Interior angles are equal]
Now in
ΔBXO and ΔDYO, we have
1 = 2...... (Alternate Interior angles)
3 = 4...... (Vertically opposite angles)
BO = OD ...... (diagonals bisects each other)
...  
ΔBXO ΔDYO   (ASA Congruence Condition)
OX = OY
  (CPCT)

 





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