Question

If two parallel lines are intersected by a transversal, then prove that bisectors of any two corresponding angles are parallel.

Answer 1

Given: AB and CD are two parallel lines and transversal EF intersects then at G and H respectively. GM and HN are the bisectors of two corresponding angles $\angle EGB$ and $\angle GHD$ respectively.

To prove: $GM\parallel HN$

Proof:
$\because AB\parallel CD$

$\therefore \angle EGB=\angle GHD$ (Corresponding angles)

$\Rightarrow \frac{1}{2}\angle EGB=\frac{1}{2}\angle GHD$

$\Rightarrow \angle 1=\angle 2$

($\angle 1$ and $\angle 2$ are the bisector of $\angle EGB$ and $\angle GHD$ respectively)

$\Rightarrow GM\parallel HN$

($\angle 1$ & $\angle 2$ are corresponding angles formed by transversal GH and GM and HN and are equal.)
Hence, proved.