Polygon

Using the degree angle formula, we have the number of degrees in one angle is:

( 6 - 2 ) $\cdot$ 180 / 6 =120

Angle ABE is a bisector of angle ABC, so 120 / 2 = $\boxed{60^\circ}$

Let  A  = (0, 2)

Let  B =  (sqrt 3 ,  1)

Let C  =  (sqrt 3  - 1)

Let D =  ( 0 , - 2)

Let  E =  (-sqrt 3 ,  -1)

AB =  sqrt  [ 3 + 1] =  2

BE = sqrt  [  (2sqrt 3)^2  + 2^2] =  sqrt  [16] =  4

AE  = sqrt  [ 3  + 3^2]  = sqrt [12]

By the Law of  Cosines

AE^2  = AB^2  + BE^2  - 2 ( AB * BE)  cos ABE

12   =  4  +  16   -  2( 2 * 4)  cos ABE

-8 / ( -16)  = cos ABE  =1/2

arccos 1/2  =   60°  =  ABE