How is the number of vertices related to the number of sides of the polygon and to the type of polygon? give two examples.

Well,

Take a triangle.  It has three vertices and three sides.  The three vertices dictate that there will be three sides.  The same rule applies for a quadrilateral (four vertices and sides), a pentagon (five vertices and sides), etc.  So the number of vertices in a polygon is equal to the number of sides in the same polygon.

Taking this a step further, since the number of vertices are equal to the number of sides, we can conclude that a polygon with three vertices is a triangle, a polygon with four vertices is a quadrilateral, a polygon with five vertices is a pentagon, etc.