Question

11. In A ABC and A 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 Fig. 8.22).
Show that
(1) quadrilateral ABED is a parallelogram
(i) quadrilateral BEFC is a parallelogram
(iiiAD II CF and AD= CF
(iv) quadrilateral ACFD is a parallelogram
zid
(v) AC=DF

(vi) A ABCEA DEF.​

Answer 1

Parallelogram :

A quadrilateral in which both pairs of opposite sides are parallel is

called a parallelogram

A

quadrilateral is a parallelogram if

i)Its

opposite sides are equal

ii)

its opposite angles are equal

iii)

diagonals bisect each other

iv)

a pair of opposite sides is equal and parallel.

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Given: ∆ABC &

∆DEF

 AB = DE and

AB || DE & BC= EF &  BC||EF

To show:

(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) ΔABC ≅ ΔDEF.

Proof:

i)

In quadrilateral ABED,

AB = DE and AB || DE

(given)

So,

quadrilateral ABED is a parallelogram

[Since a pair of opposite side is

equal and parallel]

 

(ii) In quadrilateral BEFC

Again BC = EF and

BC || EF.

so,

quadrilateral BEFC is a parallelogram.

[Since a pair of opposite side is

equal and parallel]

(iii)  Since ABED and BEFC are parallelograms.

AD = BE and BE = CF (Opposite sides of a parallelogram are equal)

Thus, AD = CF.

Also, AD || BE and BE || CF

 (Opposite

sides of a parallelogram are parallel)

Thus, AD ||CF

Hence , AD||CF & AD= CF

(iv)  AD and CF are opposite sides of quadrilateral ACFD

which are equal and parallel to each other.

 Thus, AFCD 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)

Thus, ΔABC ≅ ΔDEF (by SSS congruence rule)

 

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Hope this will help you...