Question

a transversal intersects two Parallel Lines then prove that the bisector of alternate interior angles are parallel

Answer 1

 A transversal EF intersects two lines AB and CD at M and N respectively such that ∠ 4 and ∠ 6 are a pair of alternate interior angles and ∠ 4 = ∠ 6. ... If two parallel lines areintersected by a transversal, show that the bisectors of any pair of alternate interior angles are parallel.

Answer 2

Answer:

Given :- Two parallel lines AB and CD and transversal EF intersects them at G and H respectively. GM and HN are the bisectors of the alternate angles $\angle$AGH and $\angle$GHD.

To prove :- GM ll HN.

Proof :- Since AB ll CD and transversal EF cuts them at G and H respectively. Therefore,

$\angle$AGH = $\angle$GHD

= $\sf\frac{1}{2}AGH$ = $\sf\frac{1}{2}GHD$

$\implies$ $\angle$MGH = $\angle$GHN

Thus, two lines and HN are intersected by a transversal GH at G and H respectively such that a pair of alternate angles are equal.

Hence, GM ll HN.