In a rhombus, the diagonals bisect each other at right angles and also bisect the angles of the rhombus.
Given that \( m\angle M = 74^\circ \):
Since the angles of the rhombus are bisected by the diagonals, the angle \( m\angle LOM \) will be half of \( m\angle M \):
\[ m\angle LOM = \frac{1}{2} m\angle M = \frac{1}{2} \times 74^\circ = 37^\circ \]
Thus, \( m\angle LOM \) is \( 37^\circ \).
The answer is \( 37^\circ \).