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Home / Geometry / Prove that a diagonal of a parallelogram divides it into two congruent triangles.

Prove that a diagonal of a parallelogram divides it into two congruent triangles.

General enunciation: We have to prove that a diagonal of a parallelogram divides it into two congruent triangles.

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Particular enunciation: Let the diagonal of the parallelogram ABCD is BD. The diagonal BD divides the parallelogram into two triangles ABD and BCD.

We have to prove that ∆ABD ≅ ∆BCD

Proof: Since ABCD is a parallelogram and the opposite sides and angles of a parallelogram are equal.

∴ AB = CD, BC = AD

and ∠ABC = ∠ADC

In ∆ABD and ∆BCD

AB = CD, BC = AD

and included ∠ABC = included ∠ADC

∴ ∆ABD ≅ ∆BCD.       [ side-angle-side]                                     (Proved)

 

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