if a transversal intersect two lines such that the bisectors of a pair of corresponding angles are parallel then prove that the two lines are parallel
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Step-by-step explanation:
Answer:
The transversal AD intersects the two lines PQ and RS at points B and C respectively. BE is the bisector of ∠ABQ and CG is the bisector of ∠BCS.,
and BE||CG.
We have 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), you get
1/2 ∠ ABQ = 1/2 ∠ BCS
That is , ∠ ABQ = ∠ BCS
But , they are the corresponding angles formed by the transversal AD with PQ and RS; and are equal.
Therefore, PQ || RS
(converse of corresponding angles axiom)
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