Question

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.
 Show that
(i) Quadrilateral ABED is a parallelogram
(ii) Quadrilateral BEFC is a parallelogram
(ii) AD || CF and AD=CF
(iv) Quadrilateral ACFD is a parallelogram
(v) AC=DF
(vi) △ABC≅△DEF

Answer 1

Hint: Proceed the solution of this question by keeping the properties of a parallelogram in the mind i.e. one pair of opposite sides are equal and parallel of a quadrilateral then that it will be a parallelogram. So with the help of given relation and using congruent triangle properties we have to bring those expressions so that we can prove the desired result what is asked in the question.

Complete step-by-step answer:

In this question it is given that AB=DE, AB||DE, BC=EF and BC || EF

And joining Vertices A, B and C to vertices D, E and F

So on plotting these results in the below figure

(i) Consider the quadrilateral ABED

We have , AB=DE and AB∥DE

 One pair of opposite sides are equal and parallel. Therefore

ABED is a parallelogram.


(ii) In quadrilateral BEFC , we have

⇒ BC=EF and BC∥EF. One pair of opposite sides are equal and parallel. Therefore, BEFC is a parallelogram.


(iii) As we have proved that ABED is a parallelogram

So in a parallelogram, we know that opposite sides are equal and parallel

Therefore, AD=BE and AD ∥ BE ... (1)


Similarly, we have proved that BEFC is a parallelogram

So in a parallelogram, we know that opposite sides are equal and parallel

Therefore, CF = BE and CF ∥ BE ... (2)

From (1) and (2), it can be inferred

⇒ AD=BE & CF = BE

Therefore, AD = CF


⇒ AD ∥ BE & CF ∥ BE

Therefore, AD ∥ CF


(iv) Since in above part we have proved that AD = CF and AD ∥ CF

i.e. One pair of opposite sides are equal and parallel

⇒ ACFD is a parallelogram.


(v) Since ACFD is a parallelogram. (proved in the above part)

So in a parallelogram, we know that opposite sides are equal and parallel

Therefore, AC=DF


(vi) In triangles ABC and DEF, we have

⇒ AB=DE (opposite sides of parallelogram ABED)

⇒ BC=EF (Opposite sides of parallelogram BEFC)

and CA=FD (Opposite. sides of parallelogram ACFD)


SSS (Side-Side-Side) rule-

In a triangle, if all three sides of one triangle are equivalent to the corresponding all three the sides of the other triangle, then those two triangles are said to be congruent by SSS rule
Using SSS criterion of congruence,

∴ △ABC ≅ △DEF by S.A.S axiom of congruency

Note: In this particular question, we used the condition i.e. if one pair of opposite sides are equal and parallel in a quadrilateral then it will be a parallelogram. Along with that we can also use some other property to prove it a parallelogram like

(1) If both pairs of opposite sides are parallel then it will be a parallelogram.

(2) Both pairs of opposite sides are congruent then it will be a parallelogram.

(3) If the diagonal bisects each other (that they can divide each other into two equal parts where they cross) then it will be a parallelogram.

So if we prove any above condition, then the given quadrilateral will be a parallelogram. But we should choose the condition according to what is given in question so that it can be proved.