**Solution:**

**General enunciation:** To prove that if the diagonals of a quadrilateral bisect each other, it is a parallelogram.

**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).**

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

Tags geometry mathematics parallelogram Quadrilateral

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