Question

In $△ABC$ and $△DEF,AB=DE,AB\parallel DE,BC=EF$ and $BC\parallel EF$. 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) $AD\parallel CF$ and $AD=CF$
(iv) Quadrilateral $ACFD$ is a parallelogram
(v) $AC=DF$
(vi) $△ABC\cong △DEF$

Answer 1

(i) Consider the quadrilateral $ABED$
We have , $AB=DE$ and $AB\parallel DE$

One pair of opposite sides are equal and parallel. Therefore

$ABED$ is a parallelogram.

(ii) In quadrilateral $BEFC$ , we have
$BC=EF$ and $BC\parallel EF$. One pair of opposite sides are equal and parallel.therefore ,$BEFC$ is a parallelogram.

(iii) $AD=BE$ and $AD\parallel BE$ $\mid$ As $ABED$ is a ||gm ... (1)
and $CF=BE$ and $CF\parallel BE$ $\mid$ As $BEFC$ is a ||gm ... (2)

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

$AD=CF$ and $AD\parallel CF$

(iv) $AD=CF$ and $AD\parallel CF$

One pair of opposite sides are equal and parallel

⇒ $ACFD$ is a parallelogram.

(v) Since $ACFD$ is parallelogram.

$AC=DF$ $\mid$ As Opposite sides of a|| gm $ACFD$

(vi) In triangles $ABC$ and $DEF$, we have

$AB=DE$ $\mid$ (opposite sides of $ABED$

$BC=EF$ $\mid$ (Opposite sides of $BEFC$

and $CA=FD$ $\mid$ Opposite. sides of $ACFD$

Using SSS criterion of congruence,

$△ABC\cong △DEF$