Question

Prove that the bisector of alternate interior angle x of a pair of parallel lines are parallel

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.