Question

Question 3
AP an BQ are the bisectors of the two alternate interior angles formed by the intersection of a transversal t with parallel lines l and m (in the given figure). Show that AP || BQ.

Answer 1

Given, l || m, AP and BQ are the bisectors of $\angle EABand\angle ABH$, respectively.
To prove AP || BQ.

Proof:
Since, l || m and t is transversal.
$\angle EAB=\angle ABH$ [alternate interior angles]

$\Rightarrow \frac{1}{2}\angle EAB=\frac{1}{2}\angle ABH$ [divided both sides by 2]
$\Rightarrow \angle PAB=\angle ABQ$
[AP and BQ are the bisectors of $\angle EABand\angle ABH$]
Since, $\angle PABand\angle ABQ$ are alternate interior angles angles with two lines AP and BQ and transversal AB.
Hence, AP || BQ.