Verified answer

Appropriate Question :-

In Δ ABC and Δ 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

(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) Δ ABC ≅ Δ DEF

[tex]\large\underline{\sf{Solution-}}[/tex]

Given that,

[tex]\sf \: AB \: \parallel \:DE \: and \: AB = DE \\ \\ [/tex]

We know,

In a quadrilateral, if one pair of opposite sides are equal and parallel, then quadrilateral is a parallelogram.

[tex]\sf\implies \bf \: ABED \: is \: a \: parallelogram. \\ \\ [/tex]

[tex]\sf\implies\sf \: AD\: \parallel \:BE \: \: and \: \: AD = BE - - - (1) \\ \\ [/tex]

Further given that,

[tex]\sf \: BC\: \parallel \:EF \: and \: BC = EF \\ \\ [/tex]

We know,

In a quadrilateral, if one pair of opposite sides are equal and parallel, then quadrilateral is a parallelogram.

[tex]\sf\implies \bf \: BEFC \: is \: a \: parallelogram. \\ \\ [/tex]

[tex]\sf\implies \sf \: BE\: \parallel \:CF \: \: and \: \: BE = CF - - - (2) \\ \\ [/tex]

From equation (1) and (2), we concluded that

[tex]\sf\implies \bf \: AD\: \parallel \:CF \: \: and \: \: AD = CF \\ \\ [/tex]

We know,

In a quadrilateral, if one pair of opposite sides are equal and parallel, then quadrilateral is a parallelogram.

[tex]\sf\implies \bf \: ACFD \: is \: a \: parallelogram. \\ \\ [/tex]

[tex]\sf\implies \sf \: AC\: \parallel \:DF \: \: and \: \: AC = DF \\ \\ [/tex]

Now,

[tex]\sf \: In \: \triangle \: ABC \: and \: \triangle \: DEF \\ \\ [/tex]

[tex]\qquad\sf \: AB = DE \: \: \: \: \{given \} \\ \\ [/tex]

[tex]\qquad\sf \: BC = EF \: \: \: \: \{given \} \\ \\ [/tex]

[tex]\qquad\sf \: AC = DF \: \: \: \: \{proved \: above \} \\ \\ [/tex]

[tex]\sf\implies \bf \: \triangle \: ABC \: \cong \: \triangle \: DEF \: \: \: \{SSS \: Congruency \: rule \} \\ \\ [/tex]

Answer:

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 (see Fig. 8.22). 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.

Solution:

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 (see Fig. 8.22). 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

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)

∴ ∆ABC ≅ ∆DEF

HOPE IT'S HELP YOU