Theory:

Adjacent vertices: The end-points of the same side of a polygon are called the adjacent vertices.
 
Adjacent sides: Sides of a polygon that have a common vertex are called adjacent sides. If the sides of a polygon don't have any common vertex is called non-adjacent sides.
Example:
Consider a quadrilateral \(ABCD\).
 
Th_3.1.png
 
Here, the vertex \(A\) adjacent to vertex \(B\) in the side \(AB\) and adjacent to vertex \(C\) in the side \(AC\). Thus, the vertex \(A\) is adjacent to the vertex \(B\) as well as the vertex \(C\). But the vertex \(A\) is non-adjacent to the vertex \(D\).
 
Also, the side \(AB\) is adjacent to side \(BD\) when we consider the common vertex as \(B\), and it is adjacent to the side \(AC\) when we consider the common vertex as \(A\). But the side \(AB\) is non-adjacent to the side \(AD\).
Important!
  • Convex polygons have no portions of their diagonals in their exterior which is not true in case of concave polygons.
  • For each vertex of a polygon, there are two adjacent vertices and remaining are non-adjacent vertices. Thus, in \(n\)-gon, there are '\(n-2\)' non-adjacent vertices for a vertex.
  • For each side of a polygon, there are two adjacent sides, and the remaining are non-adjacent sides. Thus, in \(n\)-gon, there are '\(n-2\)' non-adjacent sides for each side.
  • Diagonals: The line segments obtained by joining vertices which are not adjacent are called the diagonals of a polygon.
    Th_3.2.png
    The number of diagonals of a convex polygon:  If there are \(n\)-sides in a convex polygon and \(n > 3\), then it has n(n3)2 diagonals. A triangle has no diagonals.
    Example:
    Consider a \(16\)-gon.
     
    Let's find the number of diagonals in it.
     
    We have the number of sides \(n = 16\).

    Substitute the value in the formula.
    The number of diagonals in \(16\)-gon \(=\)16(163)2 \(= 104\).
    Important!
    In a triangle, all the sides/vertices are adjacent to each other. Therefore, we cannot draw diagonals in it. Thus, the number of diagonals in any triangle is \(0\).