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  • In Figure, AB || DE, AB = DE, AC || DF and AC = DF. Prove that BC || EF and BC = EF.

In Figure, AB \parallel DE, AB = DE, AC \parallel DF and AC = DF. Prove that BC \parallel EF and  BC = EF.

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Solution.
Given: AB \parallel DE  and  AC \parallel DF
Also, AB = DE and AC = DF
To prove:  BC\parallel EFand BC = EF

Proof: AB \parallel DE  and AB = DE
AC \parallel DF and AC = DF
In ACFD quadrilateral
AC\parallel FD and AC = FD   …..(1)
Thus ACFD is a parallelogram
AD\parallel CF and AD = CF …..(2)
In ABED quadrilateral
AB\parallel DE and AB = DE   …..(3)
Thus ABED is a parallelogram
AD\parallel BE  and AD = BE …..(4)
From equation 2 and 4
AD = BE = CF and AD \parallel CF \parallel BE …..(5)
In quadrilateral BCFE, BE = CF and BE\parallel CF   {from equation 5}
So, BCFE is a parallelogram BC = EF and BC\parallel EF
Hence Proved.

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