Question

If a transversal intersects two parallel lines then show that the bisectors of any pair of alternate angles are also parallel.

Answer 1

Given: $l\parallel m$




From the figure, we have,
$$\begin{gathered} \angle 1+\angle 2=\angle 3+\angle 4(\mathrm{Alternate}\mathrm{angles}).....\left(\mathrm{i}\right) \\ \mathrm{Since}\mathrm{these}\mathrm{angles}\mathrm{are}\mathrm{bisected},\mathrm{we}\mathrm{have}, \\ \angle 1=\angle 2\mathrm{and}\angle 3=\angle 4 \\ \mathrm{Substituing}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}, \\ \angle 2+\angle 2=\angle 3+\angle 3 \\ \Rightarrow 2\angle 2=2\angle 3 \\ \Rightarrow \angle 2=\angle 3 \\ \mathrm{As}\mathrm{the}\mathrm{angle}\mathrm{bisectors}\mathrm{form}\mathrm{pairs}\mathrm{of}\mathrm{alternate}\mathrm{angles},\mathrm{they}\mathrm{are}\mathrm{parallel}. \end{gathered}$$