In 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 (see figure). Show that

(i) ABED is a parallelogram

(ii) BCFE is a parallelogram

(iii) AC = DF

(iv) ∆ABC ≅ ∆DEF

Given:- AB||DE , AB=DE; BC||EF, BC=EF

Formula used:- SSS congruency rule

If all 3 sides of both triangle are equal then

Both triangles are congruent.

Solution:-

(1) In ABED

AB||DE

AB=DE

∴ ABED is a parallelogram

∵ If one pair of side in quadrilateral is equal and parallel

Then the quadrilateral is parallelogram.

(2) In BCFE

BC||EF

BC=EF

∴ BCEF is a parallelogram

∵ If one pair of side in quadrilateral is equal and parallel

Then the quadrilateral is parallelogram.

(3) As ABED is a parallelogram

BE||AD , BE=AD ;

As BCFE is a parallelogram

BE||CF , BE=CF ;

∴ By concluding both above statements

AD||CF and AD=CF

∴ ACFD is a parallelogram

∵ If one pair of side in quadrilateral is equal and parallel

Then the quadrilateral is parallelogram

If ACDF is a parallelogram ;

Then

AC=DF [opposite sides of parallelogram are equal ]

(4) In Δ ABC and Δ DEF

AB=DE [Given]

BC=EF [Given]

AC=DF [Proved above]

Δ ABC ≅ Δ DEF [SSS congruency rule]