When the mirror is rotated, the normal will turn as well, but will the incident ray and reflected ray turn?

When a mirror is rotated . . .

-- The incident ray doesn't turn.  It's just the line from the source to the mirror. 
It would be there, in the same place, even if there was no mirror.

-- The normal turns.  It's the line perpendicular to the mirror, so it must turn
with the mirror.

-- Since the normal tuns and the incident ray doesn't, the angle between them
must change.  And since the angle of the reflected ray is equal to the angle of
the incident ray, the reflected ray must also turn.