Property of Three Parallel Lines and their Transversals
Theorem :
The ratio of the intercepts made on a transversal by three parallel lines is equal to the ratio of the corresponding intercepts made on any other transversal by the same parallel lines.
Given :
line l || line m || line n and t\(_1\) and t\(_2\) are transversals.
Transversal t\(_1\) intersects the lines in points A, B, C and t\(_2\) intersects the lines in points P, Q, R.
To prove :
\(\large \frac {AB}{BC}\) = \(\large \frac {PQ}{QR}\)
Construction:
Draw seg PC, which intersects line m at point D.
Proof :
In ∆ACP,
line BD || line AP
∴ \(\large \frac {AB}{BC}\) = \(\large \frac {PD}{DC}\) … (i) [By Basic proportionality theorem]
In ∆ CPR,
line DQ || line CR
∴ \(\large \frac {PD}{DC}\) = \(\large \frac {PQ}{QR}\) … (ii) [By Basic proportionality theorem]
∴ \(\large \frac {AB}{BC}\) = \(\large \frac {PD}{DC}\) = \(\large \frac {PQ}{QR}\) … [from (i) and (ii)]
∴ \(\large \frac {AB}{BC}\) = \(\large \frac {PQ}{QR}\)
Hence proved