Question

If the bisectors of a pair of corresponding angles formed by transversal are parallel, then prove that given lines are parallel.

Answer 1

Given: AB and CD are two straight lines cut by a transversal EF at G and H respectively. GM and HN are the bisectors of corresponding angles $\angle EGB$ and $\angle GHD$ respectively such that $GM\parallel HN$.
To Prove: $AB\parallel CD$
Proof:
$\because GM\parallel HN$
$\therefore \angle 1=\angle 2$ (Corresponding angles)
$\Rightarrow 2\angle 1=2\angle 2\Rightarrow \angle EGB=\angle GHD\Rightarrow AB\parallel CD$
($\angle EGB$ & $\angle GHD$ are corresponding angles formed by transversal EF with AB and CD and are equal.)
Hence, proved.