In this lesson we will prove the basic property of parallelogram in which diagonals bisect each other.
Theorem If ABCD is a
parallelogram, then prove that the diagonals of ABCD bisect each other.
Proof
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.
Hence,

and
We will prove using
congruent triangles concept.
Consider two
Triangles ABO and COD.
1.

....( Line AC is a transversal of the parallel lines AB and CD, hence alternate angles).
2.

....(Line BD is a transversal of the parallel lines AB and CD, hence alternate angles).
3.

....(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 property.

.........(4)
From theorem that
Opposite sides of a parallelogram are equal,

..........(5)
From equation (4) and (5)
Similarly,
Hence, We conclude that AO = CO and BO = DO.
QED
To learn more about Similar and congruent triangles you can refer to
wikipedia article.
To learn more about parallelogram you can refer to
wikipedia