91Ó°ÊÓ

The angle bisector is a ray that divides the angle into two equal angles.

The theorem 4-8 states that if a point is equidistant from the sides of an angle then the point lies on the bisector of the angle.

The diagram is:

From the diagram, it can be noticed that $PX=PY$.

In thetriangles $â–³PXB$and $â–³PYB$, it can be noticed that:

$∠BXP=∠BYP=\mathbf{90}\mathbf{°}$

$BPâ‰\dots BP\left(\mathrm{common}\right)$

$PX=PY$

Therefore, it can be noticed that $BPâ‰\dots BP$,$∠BXPâ‰\dots ∠BYP$ and $PX=PY$.

Therefore, the triangles $â–³PXB$and $â–³PYB$are congruent triangles by using the HL postulate.

Therefore, by using the corresponding parts of congruent triangles it can be said that $∠PBXâ‰\dots ∠PBY$.

Therefore the $PB$ is the angle bisector of the angle $∠B$.

Therefore, if a point is equidistant from the sides of an angle then the point lies on the angle bisector.

Therefore, Theorem 4-8 is proved.