Question

If the bisectors of a pair of alternate interior angles are parallel, then prove that given lines are parallel.

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Answer 1

Given: Two lines AB and CD, PS is the transversal. Alternate angles formed by AB and CD on PS are equal.

To prove: AB║CD

Proof:

It is given that ∠BQR = ∠CRQ.    [Alternate interior angles]    .....(1)

Also, ∠CRQ + ∠QRD = 180°  [Linear pair]

⇒ ∠BQR + ∠QRD = 180°  [Using (1)]

But ∠BQR and ∠QRD forms a pair interior angles on the same side of the transversal PS and is supplementary.

Therefore, AB║CD.