Adjacent Vertices:
Adjacent vertices are the vertices that are directly connected to each other by an edge in a graph or a polygon. In other words, two vertices are considered adjacent if there is an edge that connects them. The concept of adjacent vertices is commonly used in graph theory and geometry.

Explanation:
In a graph, vertices represent points or objects, and edges represent the connections or relationships between those points. When two vertices are adjacent, it means that there is a direct connection between them, and they are considered to be neighboring points.

In the context of a polygon, adjacent vertices are the corners or points that are connected by sides. For example, in a triangle, each vertex is adjacent to two other vertices since each vertex is connected to two sides of the triangle.

Adjacent Sides:
Adjacent sides are the sides of a polygon that share a common vertex. In a polygon, each side is connected to two vertices, and if two sides share a common vertex, they are considered adjacent to each other.

Explanation:
Consider a rectangle as an example. A rectangle has four sides, and each side is adjacent to two other sides since they share two common vertices. The adjacent sides of a rectangle are always perpendicular to each other.

Similarly, in a triangle, each side is adjacent to two other sides since they share a common vertex. The concept of adjacent sides is commonly used in geometric calculations and proofs.

Visual Representation:
Here is a visual representation of adjacent vertices and adjacent sides in a triangle:

A
/ \
/ \
B-----C

In the triangle ABC, the vertices A, B, and C are adjacent to each other since they are directly connected by sides AB, BC, and CA. The sides AB, BC, and CA are adjacent to each other since they share the common vertices A, B, and C.

Conclusion:
Adjacent vertices and adjacent sides are fundamental concepts in graph theory and geometry. They describe the relationships and connections between vertices and sides in a graph or a polygon. Understanding these concepts is essential for solving problems and analyzing various structures in mathematics, computer science, and other fields.