Question

$l,m$ and $n$ are three parallel lines intersected by transversals $p$ and $q$ such that $l,m$ and $n$ cut off equal intercepts $AB$ and $BC$ on $p$. Show that $l,m$ and $n$ cut off equal intercepts $DE$ and $EF$ on $q$ also.

Answer 1

Given:$l\parallel m\parallel n$

$l,m$ and $n$ cut off equal intercepts $AB$ and $BC$ on $p$

So,$AB=BC$

To prove:$l,m$ and $n$ cut off equal intercepts $DE$ and $EF$ on $q$

i.e.,$DE=EF$

Proof:In $△ACF,$

$B$ is the mid-point of $AC$ as $AB=BC$

and $BG\parallel CF$ since $m\parallel n$

So,$G$ is the mid-point of $AF$ using line drawn throught mid-point of one side of a triangle, parallel to another side, bisects the third side.

In $△AFD,$

$G$ is the mid-point of $AF$

and $GE\parallel AD$ since $l\parallel m$

So,$E$ is the mid-point of $DF$ using line drawn throught mid-point of one side of a triangle, parallel to another side, bisects the third side.

Since $E$ is the mid-point of $DF$

$DE=EF$

Hence proved.