In triangle \( \triangle LMN \), since it is isosceles with \( \angle M \cong \angle L \) and given that \( m\angle L = 36^\circ \), it follows that:
\[ m\angle M = 36^\circ \]
Now, to find \( m\angle N \), we can use the fact that the sum of the angles in any triangle is \( 180^\circ \). Thus,
\[ m\angle L + m\angle M + m\angle N = 180^\circ \]
Substituting the known values:
\[ 36^\circ + 36^\circ + m\angle N = 180^\circ \]
This simplifies to:
\[ 72^\circ + m\angle N = 180^\circ \]
To solve for \( m\angle N \), subtract \( 72^\circ \) from \( 180^\circ \):
\[ m\angle N = 180^\circ - 72^\circ = 108^\circ \]
Therefore, the measures of the angles are:
\[ m\angle M = 36^\circ \quad \text{and} \quad m\angle N = 108^\circ \]
Final results:
- \( m\angle M = 36^\circ \)
- \( m\angle N = 108^\circ \)