In ΔABC and ΔDEF, AB=DE, AB∥DE, BC=EF and BC∥EF. Vertices A, B and C are joined to vertices D, E and F respectively. Show that (i) Quadrilateral ABED is a parallelogram (ii) Quadrilateral BEFC is a parallelogram (iii) AD∥CF and AD=CF (iv) Quadrilateral ACFD is a parallelogram (v) AC=DF (vi) ΔABC≅ΔDEF

(i) Consider the quadrilateralABEDWe have ,AB=DEandAB∥DEOne pair of opposite sides are equal and parallel. ThereforeABEDis a parallelogram (ii) In quadrilateralBEFC, we haveBC=EFandBC∥EF. One pair of opposite sides are equal and parallel.therefore ,BEFCis a parallelogram (iii)AD=BEandAD∥BE∣AsABEDis a ||gm ... (1)andCF=BEandCF∥BE∣AsBEFCis a ||gm ... (2)From (1) and (2), it can be inferredAD=CFandAD∥CF(iv)AD=CFandAD∥CFOne pair of opposite sides are equal and parallel⇒ACFDis a parallelogram (v) SinceACFDis parallelogram.AC=DF∣As Opposite sides of a|| gmACFD(vi) In trianglesABCandDEF, we haveAB=DE∣(opposite sides ofABEDBC=EF∣(Opposite sides ofBEFCandCA=FD∣Opposite. sides ofACFDUsing SSS criterion of congruence,△ABC≅△DEF