Perpendicular bisectors have a particular property: if AB is a perpendicular bisector of CD, then every point lying on AB has the same distance from C and D.
In your case, we have that every point lying on AC has the same distance from B and D.
So, in particular, we have EB=ED, because E lies on AC.
Moreover, since AC is a perpendicular bisector, it is the height of the triangle (if we choose BD as base), and it bisects BD: this means that the triangle is isosceles, so AD=AB.
This means that triangles ABE and ADE have:
- AD=AB because ABD is isosceles
- EB=ED because AC is the perpendicular bisector of BD
- AE in common
So, their sides are all equal, and thus they are congruent.