Question

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.​

Answer 1

☘️ Solution :

A transversal AD interests two lines PQ and RS at points B and C respectively. Ray BE is the bisector of ∠ABQ and ray CG is the bisector of ∠BCS; and BE || CG.

We are to prove that PQ || RS.

It is given that ray BE is the bisector of ∠ABQ.

Therefore, ∠ABE = 1/2 ∠ABQ (1)

Similarly, ray CG is the bisector of ∠BCS.

Therefore, ∠BCG = 1/2 ∠BCS. (2)

But BE || CG and AD is the transversal.

Therefore, ∠ABE = ∠BCG

(Corresponding angles axiom) (3)

Substituting (1) and (2) in (3), we get,

1/2 ∠ABQ = 1/2 ∠BCS

That is, ∠ABQ = ∠BCS

But, they are the corresponding angles formed by transversal AD with PQ and RS; and are equal.

Therefore, PQ || RS

(Converse of corresponding angles axiom)