Question

in $\Delta ABC$ and $\Delta DEF$, $AB=DE,AB||DE,BC=EF$ and $BC||EF$. Vertices$A,B$ and $C$ are joined to vertices $D,E$ and $F$ respectively. Show that
(i) quadrilateral $ABED$ is parallelogram
(ii) quadrilateral $BEFC$ is a parallelogram
(iii) $AD||CF$ and $AD=CF$
(iv) quadrilateral $ACFD$ is a parallelogram
(v) $AC=DF$
(vi) $\Delta ABC\cong \Delta DEF$

Answer 1

$\left(i\right)$ $AB=DE$ and $AB\parallel DE$ [ Given ]

One pair of opposite sides are equal and parallel to each other.

$\therefore$ $ABED$ is a parallelogram. ---- Hence proved

$\left(ii\right)$ $BC=EF$ and $BC\parallel EF$ [ Given ]

One pair of opposite sides are equal and parallel to each other.

$\therefore$ $BEFC$ is a parallelogram. ---- Hence proved.

$\left(iii\right)$ We have proved that, $ABED$ is a parallelogram.

$\Rightarrow$ So, $AD=BE$ and $AD\parallel BE$ [ Opposite sides of parallelogram are equal and parallel ] ----- ( 1 )

We also proved that, $BEFC$ is a parallelogram

$\Rightarrow$ $BE=CF$ and $BE\parallel CF$ [ Opposite sides of parallelogram are equal and parallel ] ----- ( 2 )

From ( 1 ) and ( 2 ), we get

$\Rightarrow$ $AD=CF$ and $AD\parallel CF$ --- Hence proved.

$\left(iv\right)$ Above we have proved that,

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

One pair of opposite sides are equal and parallel to each other.

$\therefore$ $ACFD$ is a parallelogram.

$\left(v\right)$ Since, $ACFD$ is a parallelogram

Then, $AC=DF$ [ Opposite sides of parallelogram are equal ]

$\left(vi\right)$ In $△ABC$ and $△DEF$

$\Rightarrow$ $AB=DE$ [ Given ]

$\Rightarrow$ $BC=EF$ [ Given ]

$\Rightarrow$ $AC=DF$ [ Proved in part (v) ]

$\therefore$ $△ABC\cong △DEF$ [ By SSS congruence rule ]