Final answer:
A rotation of 135 degrees would map a regular octagon onto itself because the octagon has rotational symmetry at multiples of 45 degrees, including 135 degrees, which aligns vertices and sides maintaining the shape.
Step-by-step explanation:
Will a rotation of 135 degrees map a regular octagon onto itself? To answer this, let's consider the symmetry of a regular octagon. A regular octagon has eight sides of equal length and eight angles of equal measure. An object has rotational symmetry if it can be rotated (less than a full turn) about a central point and still look the same.
For an octagon, this happens at multiples of a full turn (360 degrees) divided by the number of sides, which would be 360/8 = 45 degrees. Therefore, a regular octagon has rotational symmetry at 45, 90, 135, 180, ... degrees up to 360 degrees. A rotation of 135 degrees corresponds to three times 45 degrees, so it would indeed map a regular octagon onto itself.
The fact that all rotations are multiples of the smallest angle of rotation that maps the octagon onto itself ensures that vertices will align with vertices and sides with sides, thus keeping the shape's appearance unchanged.