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 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= 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.

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