## 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**