If two pairs of adjacent sides of a quadrilateral are equal, then it is a

In ∆ABC and ∆DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, Band 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) ∆ABC ≅ ∆DEF. [CBSE 2012

Given: 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) In quadrilateral ABED,AB = DE and AB || DE| Given∴ quadrilateral ABED is a parallelogram.| ∵ A quadrilateral is a parallelogram if a pair of opposite sides are paralleland are of equal length(ii) In quadrilateral BEFC,BC = EF and BC || EF | Given∴ quadrilateral BEFC is a parallelogram.| ∵ A quadrilateral is a parallelogram if a pair of opposite sides are paralleland are of equal length(iii) ∵ ABED is a parallelogram| Proved in (i)∴ AD || BE and AD = BE ...(1)| ∵ Opposite sides of a || gmare parallel and equal∵ BEFC is a parallelogram | Proved in (ii)∴ BE || CF and BE = CF ...(2)| ∵ Opposite sides of a || gmare parallel and equalFrom (1) and (2), we obtainAD || CF and AD = CF.(iv) In quadrilateral ACFD,AD || CF and AD = CF| From (iii)∴ quadrilateral ACFD is a parallelogram.| ∵ A quadrilateral is a parallelogram if a pair of opposite sides are parallel and are of equal length(v) ∵ ACFD is a parallelogram| Proved in (iv)∴ AC || DF and AC = DF.| In a parallelogram opposite sides are parallel and of equal length(vi) In ∆ABC and ∆DEF,AB = DE| ∵ ABED is a parallelogramBC = EF| ∵ BEFC is a parallelogramAC = DF | Proved in (v)∴ ∆ABC ≅ ∆DEF.

| SSS Congruence Rule

A is a quadrilateral with two pairs of adjacent sides equal.

Solution