In an isosceles triangle, the two angles opposite the equal sides are congruent. Given that \( \triangle LMN \) is isosceles with \( \angle M \cong \angle L \) and \( m\angle L = 36^\circ \), we can conclude:
\[ m\angle M = 36^\circ \]
Since the sum of the angles in any triangle is \( 180^\circ \), we can find the measure of angle \( N \) as follows:
\[ m\angle L + m\angle M + m\angle N = 180^\circ \]
Substituting the known values:
\[ 36^\circ + 36^\circ + m\angle N = 180^\circ \]
\[ 72^\circ + m\angle N = 180^\circ \]
Now, subtract \( 72^\circ \) from both sides:
\[ m\angle N = 180^\circ - 72^\circ = 108^\circ \]
Thus, the measures of the angles in triangle \( LMN \) are:
\[ m\angle M = 36^\circ, \quad m\angle L = 36^\circ, \quad m\angle N = 108^\circ. \]
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