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 $\angle ATR\mathrm{and}\angle CST$ be the corresponding angles and CM and TN respectively be their angle bisectors. Therefore
$$\begin{gathered} \angle ATN=\angle NTP=x \\ \angle CSM=\angle MST=y \end{gathered}$$
Here, SM and TN are parallel. Therefore
$\angle NTR=\angle MST\Rightarrow x=y$
Now
$$\begin{gathered} \angle CST+\angle STA=2y+\left({180}^{\circ }-2x\right) \\ ={180}^{\circ }+2y-2x \\ ={180}^{\circ }+2x-2x\left(\because x=y\right) \\ ={180}^{\circ } \end{gathered}$$
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.