Data: 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 

To Prove: (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) ∆ABC ≅ ∆DEF. 

Proof: (i) AB = DE and AB||DE (Data) 

∴ BE = AD and BE || AD 

∴ ABCD is a parallelogram. 

(ii) Similarly, BC = EF and BC || EF. 

∴ BE = CF BE || CF 

∴ BEFC is a parallelogram. 

(iii) ABED is a parallelogram. 

∴ AD = BE AD || BE ………. (i) 

Similarly, BEFC is a parallelogram. 

∴ CF = BE CF || BE ……………. (ii) 

Comparing (i) and (ii), 

AD = CF and AD || CF, 

(iv) In a quadrilateral ACFD, 

AD = CF AD || CF (proved) 

∴ AC = DF AC || DF 

∴ ACFD is a parallelogram. 

(v) ACFD is a parallelogram. 

AC = DF (opposite sides). 

(vi) In ∆ABC and ∆DEF, 

AB = DE (Data) 

BC = EF (opposite sides of a parallelogram) 

AC = DF (Opposite sides of a parallelogram) 

∴ ∆ABC ≅ ∆DEF (SSS postulate).