To determine the smallest degree measure for a regular nonagon (9-sided polygon) to rotate onto itself, we can use the formula for the angle of rotation that maps it onto itself. This angle is given by:
\[ \text{Angle} = \frac{360°}{n} \]
where \( n \) is the number of sides of the polygon. For a nonagon, \( n = 9 \).
Calculating the angle:
\[ \text{Angle} = \frac{360°}{9} = 40° \]
Thus, the smallest degree measure for a regular nonagon to rotate onto itself is 40°.