C is the incenter of isosceles triangle ABD with vertex angle ∠ABD. Does the following proof correctly justify that triangles ABC and DBC are congruent? It is given that C is the incenter of triangle ABD, so segment BC is an altitude of angle ABD.
Angles ABC and DBC are congruent according to the definition of an angle bisector.
Segments AB and DB are congruent by the definition of an isosceles triangle.
Triangles ABC and DBC share side BC, so it is congruent to itself by the reflexive property.
By the SAS postulate, triangles ABC and DBC are congruent.