Adjacent Angles
GeometryAdjacent 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. They share the same corner point (vertex) and one side, but they do not overlap.
Example
When you cut a pizza, each slice has two sides that are adjacent to the neighboring slices. Angle $ABC$ and angle $CBD$ are adjacent because they share vertex $B$ and side $BC$.
Key Insight
Adjacent means "next to." Adjacent angles share a side like neighbors share a fence. When adjacent angles together form a straight line, they are a special pair called a linear pair.
Definition
Two angles are adjacent if they share a common vertex, share exactly one common ray (their common side), and their interiors do not overlap. Adjacent angles can be complementary (sum $90$), supplementary (sum $180$), or any other sum.
Example
In a diagram where ray $OB$ lies between rays $OA$ and $OC$: angle $AOB$ and angle $BOC$ are adjacent, sharing vertex $O$ and ray $OB$. If angle $AOB = 35^\circ$ and angle $BOC = 55^\circ$, then together they form angle $AOC = 90^\circ$.
Key Insight
The angle addition postulate applies to adjacent angles: if $OB$ is in the interior of angle $AOC$, then angle $AOB$ + angle $BOC$ = angle $AOC$. This postulate is used in nearly every angle calculation in geometry.
Definition
Adjacent angles share a vertex and one bounding ray such that the interior of one does not intersect the interior of the other. Formally, angles $\alpha$ and $\beta$ are adjacent at vertex $V$ with common ray $r$ if their closures share exactly $r$ and $V$. The angle addition postulate is the measure-theoretic statement that measure is additive for adjacent angles.
Example
The angle bisector of angle $AOC$ creates two adjacent angles of equal measure. If angle $AOC = 80^\circ$, the bisector ray $OB$ creates adjacent angles $AOB$ and $BOC$ each measuring $40^\circ$.
Key Insight
Adjacency is a local property at a vertex. In the angle sum formula for polygons, each interior angle is adjacent to its two neighbors. The additivity of adjacent angles, extended inductively, gives the formula $(n-2) \times 180^\circ$ for the sum of interior angles of any $n$-gon.