In a rhombus, the diagonals bisect each other at right angles. Since diagonal \( MO \) is mentioned, angle \( MLO \) and angle \( NLO \) are opposite angles formed by the intersection of the diagonals.
We know the following properties of a rhombus:
- The diagonals bisect each other.
- The diagonals bisect the angles of the rhombus.
Given that \( m \angle M = 74^\circ \), we can find \( m \angle LOM \) using the property that the angles at the intersection of a diagonal and the respective vertices of the rhombus are equal and hence can be expressed in terms of one of the angles.
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Since \( m \angle MLO \) and \( m \angle NLO \) are the angles at the vertex opposite to angle \( M \), and angles \( MLO \) and \( NLO \) share the vertex \( O \) where the diagonals intersect: \[ m \angle M + m \angle N = 180^\circ \]
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The remaining angles in the rhombus are equal. Therefore: \[ m \angle L + m \angle O = 180^\circ. \]
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Knowing that \( m \angle M = m \angle O \) and \( m \angle L = m \angle N \): \[ m \angle L = 180^\circ - 74^\circ = 106^\circ. \]
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The diagonals bisect angle \( M \), so: \[ m \angle LOM = \frac{m \angle M}{2} = \frac{74^\circ}{2} = 37^\circ. \]
Thus, the measure of angle \( LOM \) is \( 37^\circ \).
So, \( m \angle LOM = 37^\circ \).