It is true that all angles of a polygon are formed at the vertices. But the term Vertex Angle is used in certain contexts.
If in a triangle any side is considered as the base, then the angle formed at the vertex opposite is called the vertex angle.
The vertex angle in an isosceles triangle is the angle included between the congruent sides.
In triangle ABC, AB ≅ AC. The angle formed in between these two congruent sides ∠A is the vertex angle of the triangle. It is opposite to the third side BC which is called the base of the triangle. 

Example:ABC is an isosceles triangle with AB = AC. If the measure of one of the base angles m ∠B = 65º,
find the measure of the vertex angle A.
m ∠B = m ∠C = 65º Base angles are congruent in an isosceles triangle.
m ∠A = 180  ( m ∠B + m ∠C)
= 180  (65 + 65) = 50º.
Hence the measure of the vertex angle = 50º
Vertex angles in a regular polygonA circle can be drawn circumscribing the sides of a regular polygon. The vertex angles are formed at the
circumcenter by the sides of the polygon.
In the adjoining diagram, ABCDEF is a regular Hexagon. The angles marked at the circumcenter 1, 2, 3, 4, 5 and 6 vertex angles and each angle measures = 60º. 

There are n congruent isosceles triangles formed with vertex at the center.Thus the Vertex angles of a regular polygon are congruent.Example:Find the measure of each of the vertex angles of a regular Pentagon.
The measure of an interior angle of a regular polygon of n sides is given by the formula
$\frac{(n2)180}{n}$
Hence the measure of an interior angle of a regular pentagon (5 sides) = $\frac{3\times 180}{5}$ = 108º.
Measure of each base angle of the isosceles triangle with vertex at the circumcenter =
$\frac{108}{2}$ = 54º.
Measure of the vertex angle of a regular pentagon = 180  (54 + 54) = 72º.
Alternate Method:The vertex angles of a regular polygon are congruent.
Since a regular pentagon has 5 vertex angles and angle around the center = 360º,
Measure of each vertex angle =
$\frac{360}{5}$ = 72º