In this lesson we will prove the basic property of parallelogram in which diagonals bisect each other.
If ABCD is a parallelogram
, then prove that the diagonals of ABCD bisect each other.
Let the two diagonals be AC and BD and O be the intersection point.
We have to prove that O is the midpoint of AC and also the midpoint of BD.
We will prove using congruent triangles
Consider two Triangles
ABO and COD.
....( Line AC is a transversal of the parallel lines AB and CD, hence alternate angles).
....(Line BD is a transversal of the parallel lines AB and CD, hence alternate angles).
....(Opposite angles when two lines intersect each other area equal)
From conditions 1,2 and 3
Triangle ABO is similar to triangle CDO (By Angle
-Angle similar property)
Since Triangles are similar, Hence ratio of sides are equal from similar triangles
From theorem that Opposite sides of a parallelogram are equal
From equation (4) and (5)
Hence, We conclude that AO = CO and BO = DO.
To learn more about Similar and congruent triangles you can refer to wikipedia
To learn more about parallelogram you can refer to wikipedia
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