Question

In ∆ABC and ∆DEF, AB=DE, AB||DE, BC=EF, BC||EF, vertices A, B and C are joined D to vertices D, E and F respectively. Show that:
(i) quadrilateral ABED is a parallelogram. (ii) quadrilateral BEFC is a parallelogram. (iii) AD||CF and AD-CF
(iv) ∆ABC ∆DEF​

Answer 1

Answer:

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Step-by-step explanation:

Given: In ΔABC and ΔDEF, AB = DE, AB || DE, BC = EF and BC || EF.

We can use the fact that in a quadrilateral if one pair of opposite sides are parallel and equal to each other then it will be a parallelogram. 

(i) It is given that AB = DE and AB || DE

If one pair of opposite sides of a quadrilateral are equal and parallel to each other, then it will be a parallelogram.

Therefore, quadrilateral ABED is a parallelogram.

(ii) It is given that BC = EF and BC || EF

Therefore, quadrilateral BEFC is a parallelogram.

(iii) As we had observed that ABED and BEFC are parallelograms, therefore

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

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

Thus, AD = BE = CF and AD || BE || CF

∴ AD = CF and AD || CF (Lines parallel to the same line are parallel to each other)

(iv) As we had observed that one pair of opposite sides (AD and CF) of quadrilateral ACFD are equal and parallel to each other, therefore, it is a parallelogram.

(v) As ACFD is a parallelogram, therefore, the pair of opposite sides will be equal and parallel to each other

∴ AC || DF and AC = DF

(vi) ∆ABC and ∆DEF,

AB = DE (Given)

BC = EF (Given)

AC = DF (Since ACFD is a parallelogram)

Answer 2

Answer:

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Step-by-step explanation:

(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) AD=BE and AD∥BE ∣ As ABED is a ||gm ... (1)

and CF=BE and CF∥BE ∣ As BEFC is a ||gm ... (2)

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

AD=CF and AD∥CF

(iv) AD=CF and AD∥CF

One pair of opposite sides are equal and parallel

⇒ ACFD is a parallelogram.

(v) Since ACFD is parallelogram.

AC=DF ∣ As Opposite sides of a|| gm ACFD

(vi) In triangles ABC and DEF, we have

AB=DE ∣ (opposite sides of ABED

BC=EF ∣ (Opposite sides of BEFC

and CA=FD ∣ Opposite. sides of ACFD

Using SSS criterion of congruence,

△ABC≅△DEF