Question

two line segment AC and BD bisect each other at O prove that ABCD is a parallelogram

Answer 1

AB||CD
so, AB=CD

AD||BC
so, AD=BC

ABC is a parallelogram......

Answer 2

${\underline{\tt{\red{Given-}}}}$ AC and BD are two segments bisecting each other at O.

${\underline{\tt{\red{To\:Prove-}}}}$ ABCD is a ||gm

${\underline{\tt{\red{Construction-}}}}$ Join AB, BC, CD and DA.

${\underline{\tt{\red{Proof-}}}}$ In Δs AOB and COB, we have

$\longrightarrow \tt {AO = CO}$ [Given]

$\longrightarrow \tt {BO = DO}$ [Given]

and, $\longrightarrow \tt {∠AOB = ∠COD}$ [Vertically opp. ∠s]

• So, by SAS criterion of congruence, we have

ΔAOB ≅ Δ COD

$\longrightarrow \tt { AB = CD}$ [ :- Corresponding parts of congruent triangles are equal ]

and, ∠1 = ∠2

Thus, AB and DC intersect AC at A and C respectively such that ∠1 = ∠2.

:- AB || DC

• Thus, in quadrilateral ABCD, we have

AB = DC and AB || DC

Hence, ABCD is a parallelogram.