Question

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

Answer 1

Given: $$I || m$$ and a transversal intersect these two parallel lines at $$A$$ and $$B$$ respectively.
$$AP$$ and $$BQ$$ are the bisector of two alternate angles.
To prove: $$AP || BQ$$
Proof: $$\because I || m$$ (given)
$$\Rightarrow \angle 1 = \angle 2$$
(alternate interior angles)
$$\Rightarrow \cfrac{1}{2} \angle 1 = \cfrac{1}{2} \angle 2$$
$$\Rightarrow \angle PAB = \angle QBA$$
Hence, the two lines AP and BQ are intersected by a transversal AB forming a pair of alternate angles equal.
$$\therefore$$ AP || BQ Hence proved.