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# In $$\triangle{ABC}$$ and $$\triangle{DEF}$$, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F, respectively (see figure). 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) $$\triangle{ABC}$$ $$\displaystyle \cong$$ $$\triangle{DEF}$$

Given: In $$\triangle{ABC}$$ and $$\triangle{DEF}$$, AB = DE, AB || DE, BC = EF and BC || EF

AB = DE and AB || DE ...(Given)
Since, a pair of opposite sides is equal and parallel
Therefore, ABED is a parallelogram.

BC = EF and BC || EF ...(Given)
Since, a pair of opposite sides is equal and parallel
Therefore, BEFC is a parallelogram.

(iii)Since, ABED is a parallelogram,
Also, BEFC in a parallelogram,
CF || BE and CF = BE ...(ii)
Thus, from Eq. (i) and (ii), we get,
Therefore, ACFD is a parallelogram.

(v)Since, ACFD is a parallelogram.
we get, AC = DF and AC || DF

(vi)Now, in $$\triangle{ABC}$$ and $$\triangle{DEF}$$,
AB = DE ...(Given)
BC = EF ...(Given)
and AC = DF ...(From part (v))
Therefore, $$\triangle{ABC}$$ $$\displaystyle \cong$$ $$\triangle{DEF}$$ ...(By SSS test)