Question

plz give answer of this question.☺️☺️​

Answer 1

prove that a quadrilateral is a parallelogram if its diagonals bisect each other

Proof: In ∆OPQ and ∆ORS,

OP = OR, OQ = OS (Given);

∠POQ = ∠ROS (Opposite angles).

Therefore, ∆OPQ ≅ ∆ORS (by SAS criterion of congruency)

Therefore, ∠OPQ = ∠ORS (CPCTC).

So, PQ ∥ SR (From equal alternate angles).

Similarly, from ∆OQR and ∆OSP we get PS ∥ QR.

Therefore, PQRS is a parallelogram.

Answer 2

$\tt{\huge{\purple{Question:-}}}$

$\orange{Prove\: that\: a \:quadrilateral}$

$\orange{\:is \:a\: parallelogram\: if\: the\: diagonal \:bisect \:each\: other }$

༄༄ Solution ༄༄

$\tt{\huge {\blue {\underline{\underline{Given:-}}}}}$

ABCD is a quadrilater

AB and BD diagonal bisect each other

$\therefore$ AO = OC

BO = OD

$\tt{\huge {\pink {\underline{\underline{To\:prove:-}}}}}$

ABCD is a ||gm

$\tt{\huge {\green {\underline{\underline{Solution:-}}}}}$

$In\: ∆AOB \: and \: ∆DOC \:$

$\:\:\:\:\:\:\:\:\:\: AO = OC \:\: [given]$

$\:\:\: \angle AOB = \angle DOS \:\: \\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:[vertically\:opposite\: angle]$

$\:\:\:\:\:\:\:\:\:\: AO = OC \:\: [given]$

$\:\:\:\:\:\:\:\:\:\:\: ∆AOB \:congurant\: ∆DOC [by\:SAS\: rule ]$

By C.P.C.T

$\:\:\:\:\:\:\:\:\:\:\:\: \angle OAB = \angle OCD$

But they are alternate interior angle hence AB||DC

Similarly, AD||BC

Therefore ABCD is a ||gm [Because all side of ||gm are parallel]

$\purple{\huge{\underline{\underline{Hence \:proved}}}}$