Download Title : The diagonals of a parallelogram bisect each other Objective

Title : The diagonals of a parallelogram bisect each other
Objective : In this script we will learn a Quadrilateral Theorem which states that the Diagonals of a
parallelogram bisect each other.
Let us consider a parallelogram ABCD. In this parallelogram ,let the two diagonals AC and BD
intersect at the point O .
In this theorem ,we have to prove that O is the midpoint of AC and also the midpoint of BD.
That is, we have to prove that AO = OC and BO = OD
Now, let us consider two triangles ABO and COD.
1. Angle OAB = Angle CAB = Angle ACD = Angle OCD as the line AC is a transversal of the parallel lines
AB and CD, hence they are alternate angles which are equal.
2. Angle ODC = Angle BDC = Angle DBA = Angle OBA
as the line BD is a transversal of the parallel lines AB and CD, hence they are also alternate angles which
are equal.
3. When two lines intersect each other the opposite angles are equal.
So, Angle DOC = Angle AOB as they are opposite angles
Therefore from conditions 1,2 and 3
Triangle ABO is similar to triangle CDO (By Angle -Angle similarity rule)
Since the triangles are similar, Hence the ratio of sides are equal from similar triangles property.
DC/AB =DO/OB =CO/OA.........(4)
But the opposite sides of a parallelogram are equal, therefore
DC =AB..........(5)
From equation (4) and (5)
DC/AB =DO/OB =CO/OA=1
or
DO/OB=1
therefore
DO = OB
Similarly, CO = OA
Hence, We conclude that AO = CO and BO = DO.
That is , the diagonals of a parallelogram bisect each other.
Summary : In this script we have proved that the diagonals of a parallelogram bisect each other.