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Home / Geometry / Prove that, if the diagonals of a quadrilateral bisect each other, it is a parallelogram.

Prove that, if the diagonals of a quadrilateral bisect each other, it is a parallelogram.

Solution:
General enunciation: To prove that if the diagonals of a quadrilateral bisect each other, it is a parallelogram.Screenshot_11
Particular enunciation: Let the diagonals AC and BD of the quadrilateral ABCD bisect each other at O.
We have to prove that ABCD is a parallelogram.
Proof: The diagonals AC and BD of ABCD bisect each other at O.
∴AO = CO and BO = DO
In ∆ AOB and ∆ COD,
AO = CO, BO = DO and included ∠AOB = included ∠COD [∵vertically opposite angle] ∴ ∆ AOB ≅∆ COD [by side angle side theorem] Therefore AB = CD
In ∆AOD and ∆BOC,
AO = CO, BO = DO and included ∠AOD = included ∠BOC [∵vertically opposite angle] ∴ ∆ AOD ≅∆ BOC [by side angle side theorem] Therefore AD = BC
In the quadrilateral ABCD, AB = CD and AD = BC.
∴AB||CD and AD||BC
Therefore □ABCD is a parallelogram. (proved).

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