Question

Show that the angle bisectors of a pair of alternate angles made by the transversal with two parallel lines are parallel to each other.

Answer 1

We are given two parallel lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$, and transversal $\overleftrightarrow{PQ}$. Consider the pair of alternate angles $\angle ALM$ and $\angle LMD$.

Let $\overrightarrow{LX}$ be the bisector of $\angle ALM$ and $\overrightarrow{MY}$ be the bisector of $\angle LMD$.

Extend the ray $\overrightarrow{LX}$ to the straight line $\overleftrightarrow{XR}$ and the ray $\overrightarrow{MY}$ to the straight line $\overrightarrow{SY}$ as shown in the figure(see Fig).
We have to show that $\overleftrightarrow{XR}\parallel \overleftrightarrow{SY}$. Consider the lines $\overleftrightarrow{XR}$ and $\overleftrightarrow{SY}$ with transversal $\overleftrightarrow{PQ}$.
We have,
$\angle XLM=\frac{1}{2}\angle ALM$ and $\angle LMY=\frac{1}{2}\angle LMD$.
However, $\angle ALM=\angle LMD$ ........ (Alternate angles made by same transversal $PQ$)
We hence obtain $\angle XLM=\angle LMY$.
But $\angle XLM$ and $\angle LMY$ are a pair of alternate angles made by the transversal $\overleftrightarrow{PQ}$ with the lines $\overleftrightarrow{XR}$ and $\overleftrightarrow{SY}$.
Hence, we conclude that $\overleftrightarrow{XR}\parallel \overleftrightarrow{SY}$