Is triangle ADB congruent to triangle CDB by SAS, AAS, ASA, SSS, or HL?
If segment AB is congruent to segment CB and segment BD bisects segment AC.
We are given `Delta ABC` with `bar(AB) cong bar(CB)` and `bar(BD)` bisects `bar(AC)` at `D` .
(1) Since AB=CB we know that angles A and C are congruent from the isosceles triangle theorem. Also since `bar(AC)` is bisected at D, we have AD=CD so the `Delta ADB cong Delta CDB` by SAS
(2) BD=BD by the reflexive property. Since AD=CD you could use SSS.
(3) If you know that the median drawn from the vertex angle of an isosceles triangle is also the angle bisector of the vertex angle, the altitude drawn from the vertex angle, and the perpendicular bisector of the base then you could use any of AAS,ASA, or H-L.