adjacent angles
Two angles are adjacent when they sit right next to each other and share a wall. Picture a door swung partway open and then a second door hinged on the first door's edge: the two openings sit side by side, sharing the middle edge, with no overlap. Adjacent angles are the geometric version of two pie slices cut from the same point that share one straight cut between them.
Precisely: angle ABC and angle CBD are adjacent if they share the vertex B and the common side ray BC, while their other sides (ray BA and ray BD) lie on opposite sides of that common ray, and the angles do not overlap. Because they fit together edge to edge, their measures simply add: by the angle-addition postulate, m(angle ABC) + m(angle CBD) = m(angle ABD). That additivity is exactly what makes adjacent angles useful — you can split a big angle into two adjacent pieces, or build a big angle from two smaller adjacent ones.
A caution: adjacency is about position, not about any special sum. Adjacent angles need not add to 90 or 180; they add to whatever the outer angle happens to be. When two adjacent angles do happen to form a straight line, they are a special case called a linear pair, and only then do they sum to 180.
Ray BC splits angle ABD into angle ABC = 40 degrees and angle CBD = 25 degrees. These are adjacent angles sharing side BC, so the whole angle ABD measures 40 + 25 = 65 degrees.
Adjacent angles share a vertex and a side, and their measures add.
Two angles that merely have the same measure, or that are across a crossing, are not adjacent — adjacency requires a shared vertex and a shared side with no overlap.