Question

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

Answer 1

Proof:
let the angle be ABC (B is the vertex with the angle). Let its angle bisector be BD. Assume a point E anywhere on the line BD. Join E to BC and BA PERPENDICULARLY because distance is always measured perpendicularly. Now we have two right triangles BAE and BCE.
Now in these two triangles-
Statements Reasons

Angle ABE = Angle CBE & & & & & & & & & & & & & & & & ( angle bisector)
Side BE = Side BE & & & & & & & & & & & & & & & & & & & & & (common line to both the triangles)
Angle BAE = Angle BCE & (right angles)(90*)

Hence by Angle-side-angle CONGRUENCE rule, triangle BAE is CONGRUENT TO triangle BCE.
and by CPCT AE IS EQUAL TO CE. (which is the perpendicular distance).