Question

If in the figure, the bisectors $ AP$ and $ BQ$ of the alternate interior angles are parallel, then show that $ l\left|\right|m$.

Answer 1

Step 1: State the given data and equate $\angle PAB$ to $\angle ABQ$

Let us draw and label the diagram as follow.

It is given that $AP\left|\right|BQ$.

And, $AP$ and $BQ$ are the bisectors of the alternate interior angles.

So, $\angle 1=\angle 2$ $...\left(\mathrm{i}\right)$

And $\angle 3=\angle 4$ $...\left(\mathrm{ii}\right)$

Now, since $AP\left|\right|BQ$ and $AB$ is a transversal.

So, $\angle PAB=\angle ABQ$ [alternate interior angles]

$\Rightarrow$ $\angle 2=\angle 3$

Step 2: Show that $l$ is parallel to $m$:

Now, adding equation $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$, we get,

$\angle 1+\angle 3=\angle 2+\angle 4$

$\Rightarrow$ $\angle 1+\angle 2=\angle 2+\angle 4$ $\left[\because \angle 2=\angle 3\right]$

$\Rightarrow$ $\angle 1+\angle 2=\angle 3+\angle 4$ $\left[\because \angle 2=\angle 3\right]$

$\Rightarrow$ $\angle MAB=\angle ABS$ [from the diagram]

Now, since $\angle MAB$ and $\angle ABS$ are alternate interior angles that are equal.

Then, by the rules of alternate interior angles line $l$ must be parallel to line $m$.

Hence, it is proved that $l\left|\right|m$.