Question

In the follow diagram, lines l, m and n are parallel to each other. Two transversals p and q intersect the parallel lines l, m and n at points A,B,C and P,Q,R respectively as shown below. Prove that $\frac{AB}{BC}=\frac{PQ}{QR}$ .

Answer 1

$\text{In the given figure,}l||m||n$

Construction: Join AR which intersects BQ at D.

$\text{In}\Delta ACR$
$BD||CR(\because m||n)$

$\frac{AB}{BC}=\frac{AD}{DR}...\left(i\right)$
$\text{(By basic proportionality theorem)}$

$\text{Similarly in}\Delta ARP$,

$DQ||AP(\because l||m)$

$\frac{PQ}{QR}=\frac{AD}{DR}...\left(ii\right)$
$\text{(By basic proportionality theorem)}$

$\text{From (i) and (ii),}$

$\frac{AB}{BC}=\frac{PQ}{QR}$