Complementary angles are two angles whose measures add up to 90 degrees.
Given that the measure of angle \( m \angle 2 = 85^\circ \), we can find the measure of angle \( m \angle 1 \) using the following equation:
\[ m \angle 1 + m \angle 2 = 90^\circ \]
Substituting the value of \( m \angle 2 \):
\[ m \angle 1 + 85^\circ = 90^\circ \]
Now, solve for \( m \angle 1 \):
\[ m \angle 1 = 90^\circ - 85^\circ = 5^\circ \]
Therefore, the value of \( m \angle 1 \) is \( 5^\circ \).