Given: l || m and a transversal intersect these two parallel lines at A and B respectively.

AP and BQ are the bisectors of two alternate angles

**To prove:** AP || BQ

**Proof: **∵ l || m (given)

⇒ ∠1 = ∠2

(alternate interior angles)

⇒ \(\frac { 1 }{ 2 }\)∠1 = \(\frac { 1 }{ 2 }\)∠2

⇒ ∠PAB = ∠QBA

Hence, the two lines AP and BQ are intersected by a transversal AB forming a pair of alternate angles equal.

∴AP || BQ Hence proved