Question

If a transversal intersect two lines such that the bisecters of a pair of alternate interior angles are parallel, then prove that the two lines are parallel ​

Answer 1

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= 1/2∠ABQ

In the same way,

∠BCF= 1/2∠BCS

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

∠ABE=∠BCF

1/2∠ABQ=1/2 ∠BCS

∠ABQ=∠BCS

Therefore, by the converse of corresponding angle axiom,

PQ∥RS.