Question

If three or more parallel lines are intersected by two transversals prove that the intercepts made by them on the transversals are proportional

Answer 1

Three parallel lines $l,m,n$ which are cut by the transversals $AB$ and $CD$ in $P,Q,R$ and $E,F,G$ respectively

We shall prove,$\frac{PQ}{QR}=\frac{EF}{FG}$

Draw $PL\parallel CD$ meeting the lines $m$and $n$ in $M$ and $L$ respectively

PROOF: Since $PE\parallel MF$ and $PM\parallel EF$

$\therefore$ $PMFE$ is a parallelogram

$\Rightarrow PM=EF$ ....(i) (Opposite sides of a parallelogram are equal )

Also $MF\parallel n$ and $ML\parallel FG$

$\therefore MLGF$ is a parallelogram and $ML=FG$ (ii) (Opposite side of a ${\parallel }^{gm}$ are equal )

Now in $△PRL$ ,

$QM$ is parallel to $RL$

$\frac{PQ}{QR}=\frac{PM}{ML}$ [Using Basic proportionality Theorem]

Thus, $\frac{PQ}{QR}=\frac{EF}{FG}$