Question

0. 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.
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Answer 1

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❄As per the information provided, let's say that a transversal AD intersects two lines PQ and RS at points B and C respectively. Ray BE is the bisector of angle ABQ and ray CG is the bisector of angle BCS; and BE || CG.

We are to prove that PQ || RS.

[Have a glimpse at the attachment above!]

Now,

It's given that ray BE is the bisector of angle ABQ.

Therefore,

angle ABE = 1/2 angle ABQ. ....→ (i)

Similarly, ray CG is the bisector of angle BCS.

Therefore,

angle BCG = 1/2 angle BCS. ....→ (ii)

But, BE || CG and AD is the transversal.

Therefore,

angle ABE = angle BCG

(Corresponding angles axiom) ....→ (iii)

Now,

On substituting (i) and (ii) in (iii), we get,

1/2 angle ABQ = 1/2 angle BCS.

That is,

angle ABQ = angle 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)

✬_____Hence Proved_____✬

Answer 2

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