A regular octagon is mapped onto itself every time it is rotatedA) 45° B) 60° C) 85° D) 108°

A REGULAR octagon has 8 sides. Being regular, all of its internal angles and sides are equal to each other. Let's focus on the picture below. The 2 lines in blue are perpendicular, and the red one, being equally spaced to both of them, is at 90/2=45º from each. 
As such, if you rotate the octagon 45 degrees it will fit in the same place, it will also fit if you rotate multiples of 45, such as 90, 135, 180, 225, etc.
Hence, the only solution available in the options is A) 45º