Adjacent Angles
Adjacent angles are two angles that share a common vertex and a common side but do not overlap.
Definition
Adjacent angles are two angles that sit next to each other, sharing the same vertex and exactly one common side (ray), without their interiors overlapping. They can be complementary, supplementary, or any other sum, and the angle addition postulate applies to them: if ray $OB$ lies in the interior of angle $AOC$, then angle $AOB$ + angle $BOC$ = angle $AOC$.
Example
When you cut a pizza, each slice has two sides adjacent to its neighboring slices; angle $ABC$ and angle $CBD$ are adjacent because they share vertex $B$ and side $BC$. If angle $AOB = 35^\circ$ and angle $BOC = 55^\circ$ with $OB$ between rays $OA$ and $OC$, together they form angle $AOC = 90^\circ$. The angle bisector of angle $AOC = 80^\circ$ creates two adjacent angles of $40^\circ$ each.
Key Insight
Adjacent means "next to": adjacent angles share a side like neighbors share a fence, and when they together form a straight line, they are called a linear pair. Adjacency is a local property at a vertex, and extending the angle addition postulate inductively around every vertex of a polygon gives the formula $(n-2) \times 180^\circ$ for the sum of interior angles of any $n$-gon.