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If a transversal intersects two parallel lines l and m then prove that the bisectors AP and BQ of any two alternate angles are parallel i.e. AP || BQ.

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Given: l || m and a transversal intersect these two parallel lines at A and B respectively.

AP and BQ are the bisectors of two alternate angles

To prove: AP || BQ

Proof: ∵ l || m (given)

⇒ ∠1 = ∠2

(alternate interior angles)

⇒ \(\frac { 1 }{ 2 }\)∠1 = \(\frac { 1 }{ 2 }\)∠2

⇒ ∠PAB = ∠QBA

Hence, the two lines AP and BQ are intersected by a transversal AB forming a pair of alternate angles equal.

∴AP || BQ Hence proved

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