In a quadrilateral ABCD, a line segment BD is a diagonal such that AB=CD and ∠ABD=∠CDB. Are the triangles ABD and CDB congruent? If so, by what rule?

Correct option is B

Given:
AB = CD
∠ABD = ∠CDB
BD is common to both triangles
Formula Used:
SAS (Side–Angle–Side) Congruence Rule:
If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.
Solution:
Consider triangles ABD and CDB:
AB = CD (Given)
BD = BD (Common side)
∠ABD = ∠CDB (Given)
Thus, two sides and the included angle of triangle ABD are equal to the corresponding parts of triangle CDB.
Therefore, ΔABD ≅ ΔCDB (by SAS rule).
Correct Option: B — Yes, by SAS