Question

Prove logically the diagonals of a parallelogram bisect each other. Show that conversely that a quadrilateral in which diagonals bisect each other is a parallelogram.

Answer 1

Hint: In order to prove the above statement, we will use the properties of parallelogram which are mentioned below. Along with that we will also use alternate angles property, concept of congruent triangles to solve the converse of the statement.
Properties of parallelogram are-
$ \to $Opposite sides are parallel
$ \to $Opposite sides are equal
$ \to $Opposite angles are equal.



Complete step-by-step answer:

Proof- Draw a parallelogram ABCD (Refer figure 1 given below)
Construction- Draw both of its diagonals AC and BD to intersect at a point O.
In $\vartriangle $ OAB and $\vartriangle $ OCD
Mark the angles formed as $\angle $ 1, $\angle $2, $\angle $3, $\angle $4
$ \Rightarrow $ $\angle $1 = $\angle $3
Or $\angle $2 = $\angle $4
{AB//CD and AC transversal and interior alternate angles}
$ \Rightarrow $AB = CD {By using the property of parallelogram}
 Now, we get
$\vartriangle $ OCD is congruent to $\vartriangle $ OAB
{By using angle side angle property,
CO = OA and DO = OB
$\therefore $ Diagonals bisect each other.

Converse- If the diagonals of a quadrilateral bisect each other then it is a parallelogram.
Proof- ABCD is a parallelogram (Refer figure 2 given below)
AC and BD are the diagonals bisect each other at O such that
Or OB = OD
In $\vartriangle $ AOB and $\vartriangle $ COD,
OB = OD
Similarly,
OA = OC
$\angle $ AOB = $\angle $COD
$\therefore $ $\vartriangle $ AOB is congruent to $\vartriangle $ COD
$ \Rightarrow $ AB = CD [C.P.C.T]
$ \Rightarrow $ $\angle $ OAB = $\angle $OCA [Alternate angles]
AB parallel CD
Similarly,
AD = BC, AD parallel BC
$\therefore $ ABCD is a parallelogram.

Note- Here one can get confused with the concept of congruent triangles which we used in the above method, so we should understand that congruent triangles will give us the corresponding parts of congruent triangles such that it will help to prove the above mentioned statement.