You can show this using the HA congruence theorem for right triangles. The distance of interest is the perpendicular distance from the point to the side of the angle. The distance from the point to the vertex is the same for both triangles (reflexive property), and the vertex angles are the same (definition of angle bisector). Hence the conditions required for HA congruence are met.
The distances from the point to the sides are then corresponding parts of congruent triangles, so are congruent. That is, the distance is the same, so <em>the point is equidistant from the sides</em>.
