Question: are the triangles in a rhombus congruent?
Answer: A rhombus is by definition a quadrilateral all four of whose sides are equal. In your rhombus, triangles ABD and CBD are congruent, since they have side BD in common and sides AB= CB and AD = CD. So in particular angles ABD and CBD are equal. ... Conversely, a parallelogram whose diagonals are perpendicular is a rhombus.