"Question 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 the given 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) ΔABC ≅ ΔDEF.
Class 9 - Math - Quadrilaterals Page 147"
Answers
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...
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