M and N are points on opposite sides AD and BC of a parallelogram ABCD such that MN passes through the point of intersection O

Question:

M and N are points on opposite sides AD and BC of a parallelogram ABCD such that MN passes through the point of intersection O of its diagonals AC and BD. Show that MN is bisected at O.

 

Solution:

Given: A parallelogram ABCD
 
To prove: MN is bisected at O

Proof:

In  ">ΔOAM and ">ΔOCN,

OA = OC                 (Diagonals of parallelogram bisect each other)

AOM = CON      (Vertically opposite angles)

MAO = OCN       (Alternate interior angles)

"> By ASA congruence criteria,

$\Delta O A M \cong \Delta O C N$

$\Rightarrow O M=O N(\mathrm{CPCT})$

Hence, MN is bisected at O.

 

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