If the sum of the interior angles of a polygon equals 1080 degree's what is the measure of each interior angle of the polygon?

Sum of interior angles = (n-2)*180 degrees = 1080 deg

So (n-2) = 1080/180 = 6 => n = 8.

The polygon is, therefore, an octagon. However, there is no reason to assume that the interior angles of this polygon are all the same - they could all be different with the only constraint being their sum.

IF, and that is a big if, the polygon were regular, then all its angles would be equal and each interior angle = 1080/8 = 135 degrees.