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).