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

Answer:

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.

Step-by-step explanation:

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Answer 2

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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.