In triangle \( \triangle LMN \), since it is given that \( \angle M \cong \angle L \) and \( m\angle L = 36^\circ \), we can determine the measures of the angles.
Since \( \triangle LMN \) is isosceles with \( \angle M \cong \angle L \), we have: \[ m\angle M = m\angle L = 36^\circ \]
To find the measure of angle \( N \), we can use the fact that the sum of the angles in a triangle is always \( 180^\circ \): \[ m\angle L + m\angle M + m\angle N = 180^\circ \]
Substituting the known values into the equation: \[ 36^\circ + 36^\circ + m\angle N = 180^\circ \]
Combining the angles: \[ 72^\circ + m\angle N = 180^\circ \]
Now, isolate \( m\angle N \): \[ m\angle N = 180^\circ - 72^\circ = 108^\circ \]
So, the measures of the angles are: \[ m\angle M = 36^\circ, \quad m\angle L = 36^\circ, \quad m\angle N = 108^\circ \]