Question

Prove the diagonals of parallelogram divide it into two congurent triangles.

Answer 1

Theorem 1 - The Diagonal Divides a Parallelogram into Two Congruent Triangles .
Given: A parallelogram ABCD.
To Prove: Δ ABD ≅ Δ DCB
Construction: Join BD
Proof :
Since ABCD is a parallelogram . AB|| CD and BD is the transversal.
In Δ ABD and Δ DCB,
∠ ABD = ∠ CDB ( Alternate interior angles )
Since AD||BC
∠ ADB = ∠ CBD …… (Alternate interior angles)
AB = CD ( opposite sides of a parallelogram)
AD = CB ( opposite sides of a parallelogram)
BD = DB ( common )
∴ Δ ABD ≅ Δ DCB (By SAS)
∴ Diagonal BD divides parallelogram ABCD into two congruent triangles ABD and DCB
Similarly diagonal AC divides || ABCD into two congruent triangles ABC and ADC.
Hence the diagonal divides a parallelogram into two congruent triangles .
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Answer 2

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