Vertex of an Angle
The vertex of an angle is the common endpoint shared by the two rays that form the angle.
Definition
The vertex of an angle is the corner point where its two sides (rays) meet; every angle has exactly one vertex, and in angle notation it is always the middle letter, so in angle $ABC$, $B$ is the vertex. Polygons have vertices at each corner where two sides meet, and a polygon with $n$ sides has $n$ vertices and $n$ interior angles. In polyhedral geometry, a vertex is a point where three or more edges meet, characterized as an extreme point of the convex hull, and in graph theory the term generalizes further to any node of a graph.
Example
Triangle $DEF$ has three angles and three vertices: $D$, $E$, and $F$; at vertex $D$, sides $DE$ and $DF$ form angle $D$, and the sum of all three angles equals $180^\circ$. A convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces satisfies Euler's formula $V - E + F = 2$: a cube has $8$ vertices, $12$ edges, and $6$ faces, so $8 - 12 + 6 = 2$.
Key Insight
The word "vertex" comes from Latin meaning "highest point" or "turning point," and in a polygon the number of vertices always equals the number of sides and the number of angles. The concept unifies across dimensions, a point itself in 0D, an endpoint of a segment in 1D, a corner of a polygon in 2D, a corner of a polyhedron in 3D, and Euler's formula connecting vertices, edges, and faces is a topological invariant true for any convex polyhedron.