The diagonals of a quadrilateral bisect each other.
Prove that this quadrilateral as a parallelogram.

Question

The diagonals of a quadrilateral bisect each other.Prove that this quadrilateral as a parallelogram.

Philoid · Accepted Answer

Given here a quadrilateral of sides MN , NK, KL and ML

Also it is given that the diagonals of the quadrilateral bisect each other.

To prove: Quad KLMN is a parallelogram

Proof: As the diagonals bisect each other,

So, NO=OL and KO=OM……………..(1)

Now, in ∆KOL and ∆NOM,

∠KOL=∠NOM (vertically opposite angles)

KO=OM (from (1))

NO=OL (from (1))

∴ ∆KOL ∆NOM (by SAS rule)

So, KL=NM (by cpct)

∠OMN=∠OKL (by cpct)………..(2)

∠OLK=∠ONM (by cpct)…………(3)

Similarly, ∆KON ∆LOM (by SAS rule)

So, KN=LM (by cpct)

∠ONK=∠OLM (by cpct)………..(4)

∠OKN=∠OML (by cpct)…………(5)

From (2) and (3) ,

We can say they are alternate interior angles.

So, KL||NM

Similarly, From (4) and (5) ,

We can say they are alternate interior angles.

So, KN||ML

Hence, KLMN is a parallelogram.