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In ∆ ABC and ∆ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (see Fig.). Show that 

(i) quadrilateral ABED is a parallelogram 

(ii) quadrilataeral BEFC is a parallelogram 

(iii) AD || CF and AD = CF 

(iv) quadrilateral ACFD is a parallelogram 

(v) AC = DF 

(vi) ∆ ABC ≡ ∆ DEF

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Given : In DABC and DDEF, AB = DE, AB ||DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F. 

To Prove : (i) ABED is a parallelogram 

(ii) BEFC is a parallelogram 

(iii) AD || CF and AD = CF 

(iv) ACFD is a parallelogram 

(v) AC = DF 

(vi) ∆ABC ≅ ∆DEF 

Proof : (i) In quadrilateral ABED, we have 

AB = DE and AB || DE. [Given] 

⇒ ABED is a parallelogram. [One pair of opposite sides is parallel and equal] 

(ii) In quadrilateral BEFC, we have 

BC = EF and BC || EF [Given] 

⇒ BEFC is a parallelogram. [One pair of opposite sides is parallel and equal] 

(iii) BE = CF and BE||BECF [BEFC is parallelogram] 

AD = BE and AD||BE [ABED is a parallelogram] 

⇒ AD = CF and AD||CF 

(iv) ACFD is a parallelogram. [One pair of opposite sides is parallel and equal] 

(v) AC = DF [Opposite sides of parallelogram ACFD] 

(vi) In ∆ABC and ∆DEF, we have 

AB = DE [Given] 

BC = EF [Given] 

AC = DF [Proved above] 

∴ ∆ABC ≅ ∆DEF [SSS axiom]

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