Let's sketch a few sides of the regular polygon and extend
two alternate sides until they meet at a right angle ∠C.




Suppose the regular polygon has n sides.

∠DAB ≅ ∠EBA because interior angles of 
a regular polygon are congruent.

∠CAB ≅ ∠CBA  because they are supplements of
congruent angles

∠ACB is a right angle because it is given that
the extensions of DA and EB form a right angle.

ΔABC is an isosceles right triangle.

m∠CAB = m∠CBA = 45° because they are base angles
of an isosceles right triangle. 

∠CAB, which has measure 45°, is an exterior angle of the polygon.

The sum of all n exterior angles of any polygon is 360°

All n exterior angles of a regular polygon are congruent.

Therefore 

n×45° = 360°
    n = 
    n = 8

So the regular polygon has 8 sides.  It is a regular octagon:



Edwin