Question

If a transversal intersects two lines such that the bisectors of a pair of alternate interior angles

are parallel, then prove that the two lines are parallel.​

Answer 1

Hope it helps you.

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=
2
1
​
∠ABQ
In the same way,
∠BCF=
2
1
​
∠BCS
Since BE and CF are parallel and AD is the transversal, therefore, by corresponding angle axiom,
∠ABE=∠BCF
2
1
​
∠ABQ=
2
1
​
∠BCS
∠ABQ=∠BCS

Therefore, by the converse of corresponding angle axiom,
PQ||RS


The bottom most point of the transversal is D.