91Ó°ÊÓ

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

The theorem 4-7 states that if a point lies on the bisector of an angle then the point is equidistant from the sides of the angle.

The diagram is:

From the diagram, it can be noticed that $PB$ is the angle bisector of the angle $∠B$and the point $p$ is lying on the angle bisector.

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

role="math" localid="1649231285910" $∠XBP=∠YBP(âˆ\mu PB\mathbf{is}\mathbf{}\mathbf{angle}\mathbf{}\mathbf{bisector})$

$∠BXP=∠BYP=\mathbf{90}\mathbf{°}$

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

Therefore, it can be noticed that $∠XBPâ‰\dots ∠YBP$,$∠BXPâ‰\dots ∠BYP$and $BPâ‰\dots BP$.

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

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

Therefore the point$P$ is equidistant from the sides of the angle.

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

Therefore, Theorem 4-7 is proved.