Question

D is the midpoint of AC, BDC=BDA prove ABD=CBD

Answer 1

In triangle ABD and CDB
AD=DC (Since D is the midpoint)
BD= BD (common side)
and
angle BDA= angle BDC
so by SAS (side angle side) the two triangles are congruent
using CPCTC angle ABD= angle CBD

Answer 2

By inspection, line AC has to be normal to line BD in order for the angles in question to be equal.\[\text{ABD}=\text{ArcTan }\left[\frac{\text{AD}}{\text{BD}}\right],\text{DBC}=\text{ArcTan }\left[\frac{\text{DC}}{\text{BD}}\right] \]AD=DC, so the angles ABD and DBC are equal to each other.