11. 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 (see Fig. 8.22). Show that (i) quadrilateral ABED is a parallelogram (ii) quadrilateral BEFC is a parallelogram (iii) AD || CF and AD = CF
Solution: (i) AB = DE and AB || DE (Given) Two opposite sides of a quadrilateral are equal and parallel to each other. Thus, quadrilateral ABED is a parallelogram (ii) Again BC = EF and BC || EF. Thus, quadrilateral BEFC is a parallelogram. (iii) Since ABED and BEFC are parallelograms. ⇒ AD = BE and BE = CF (Opposite sides of a parallelogram are equal) , AD = CF. Also, AD || BE and BE || CF (Opposite sides of a parallelogram are parallel) , AD || CF (iv) AD and CF are opposite sides of quadrilateral ACFD which are equal and parallel to each other. Thus, it is a parallelogram. (v) Since ACFD is a parallelogram AC || DF and AC = DF (vi) In ΔABC and ΔDEF, AB = DE (Given) BC = EF (Given) AC = DF (Opposite sides of a parallelogram) , ΔABC ≅ ΔDEF [SSS congruency]
