Question

If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

Answer 1

Let m and l be two lines and p be the transversal line.

Let $\angle ATR$ and $\angle CST$ be the corresponding angles and CM and TN respectively be their angle bisectors. Therefore
$\angle ATN$=$\angle NTP$=$x$

$\angle CSM$=$\angle MST$=$y$
Here, SM and TN are parallel. Therefore
$\angle NTR=\angle MST\Rightarrow x=y$
Now
$\angle CST+\angle STA=2y+({180}^{\circ }-2x)$

$={180}^{\circ }+2y-2x$ $={180}^{\circ }+2x-2x$ ($\because x=y$)

$={180}^{\circ }$

Thus, the sum of the interior angles on the same side of the line p and between the lines l and m is ${180}^{\circ }$.
Therefore, l and m are parallel.
Hence, if a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then the two lines are parallel.