A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel.

Prove that through a given point, we can draw only one perpendicular to a given line.

In the given fig., $\angle \mathrm{Q}>\angle \mathrm{R}, \mathrm{PA}$ is the bisector of $\angle \mathrm{QPR}$ and  $PM\perp QR$. Prove that $\angle \mathrm{APM}=\frac{1}{2}(\angle \mathrm{Q}-\angle \mathrm{R})$.