If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel

The transversal AD intersects the two lines PQ and RS at points B and C respectively. BE is the bisector of ∠ABQ and CF is the bisector of ∠BCS.

As, BE is the bisector of ∠ABQ, then,

∠ABE=21​∠ABQ

In the same way,

∠BCF=21​∠BCS

Since BE and CF are parallel and AD is the transversal, therefore, by corresponding angle axiom,

∠ABE=∠BCF

21​∠ABQ=21​∠BCS

∠ABQ=∠BCS

Therefore, by the converse of corresponding angle axiom,

PQ∥RS.