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