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
(ii) quadrilateral BEFC is a parallelogram
(ii) AD || CF and AD=CF
(iv) quadrilateral ACFD is a parallelogram
(v) AC=DF
(vi) A ABC = A DEF
Fig. 8.22​

Answer 1

$\sf Appropriate \: Question:$

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. 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

$\large\underline{\sf{Solution-}}$

Given that,

$\sf \: AB \: \parallel \:DE \: and \: AB = DE$

We know,

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

$\sf\implies \bf \: ABED \: is \: a \: parallelogram.$

$\sf\implies\sf \: AD\: \parallel \:BE \: \: and \: \: AD = BE - - - (1)$

Further given that,

$\sf \: BC\: \parallel \:EF \: and \: BC = EF$

We know,

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

$\sf\implies \bf \: BEFC \: is \: a \: parallelogram.$

$\sf\implies \sf \: BE\: \parallel \:CF \: \: and \: \: BE = CF - - - (2)$

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

$\sf\implies \bf \: AD\: \parallel \:CF \: \: and \: \: AD = CF$

We know,

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

$\sf\implies \bf \: ACFD \: is \: a \: parallelogram.$

$\sf\implies \sf \: AC\: \parallel \:DF \: \: and \: \: AC = DF$

Now,

$\sf \: In \: \triangle \: ABC \: and \: \triangle \: DEF$

$\qquad\sf \: AB = DE \: \: \: \: \{given \}$

$\qquad\sf \: BC = EF \: \: \: \: \{given \}$

$\qquad\sf \: AC = DF \: \: \: \: \{proved \: above \}$

$\sf\implies \bf \: \triangle \: ABC \: \cong \: \triangle \: DEF \: \: \: \{SSS \: Congruency \: rule \}$