Adjacent angles are two angles in a plane that have a common side and a common vertex.
Adjacent angles do not share points in their interior. In other words, they do not overlap.
Therefore, you could not say that angle ABC and angle DBC are adjacent. Angle ABC and angle DBC have a common vertex ( vertex B) and a common side ( side BC ). However, they overlap since angle DBC is included in angle ABC.
Angle ABC and angle DCB are not adjacent. Although they have a common side that is BC, they do not have a common vertex.
The vertex of angle ABC is B and the vertex of angle DCB is C.
For the figure below, angle AED and angle BEC are not adjacent angles. Although they have a common vertex that is E, they have no common side.