Let the number of sides (and angles) of the polygon be n

The formula for the the sum S of the n interior angles
of an n-sided polygon is:

S = (n - 2)*180°.

Since the polygon is regular, all its n interior angles
are the same.  Therefore the sum of them is (135°) times n,
or 135n°

So we also have S = 135n°

So setting equals equal:

(n - 2)*180° = 135n°

(n - 2)*180 = 135n

180(n - 2) = 135n

180n - 360 = 135n

180n - 135n = 360

45n = 360

  n = 8

So the regular polygon has 8 sides and is
therefore an octagon.

Edwin