# In ∆ ABC and ∆ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E

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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

(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

(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]

(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]