Lesson Proof: The diagonals of parallelogram bisect each other

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