In an isosceles triangle, two angles are equal. Since triangle \( \triangle LMN \) is isosceles with \( \angle M \cong \angle L \) and \( m\angle L = 36^\circ \), we can also conclude that:
\[ m\angle M = m\angle L = 36^\circ \]
Now, to find the measure of angle \( N \), we can use the fact that the sum of the interior angles of any triangle is \( 180^\circ \). Therefore, we have:
\[ m\angle L + m\angle M + m\angle N = 180^\circ \]
Substituting the known values:
\[ 36^\circ + 36^\circ + m\angle N = 180^\circ \]
Combining the measures of angles \( L \) and \( M \):
\[ 72^\circ + m\angle N = 180^\circ \]
Now, we can solve for \( m\angle N \):
\[ m\angle N = 180^\circ - 72^\circ = 108^\circ \]
Thus, the measures of the angles in triangle \( \triangle LMN \) are:
- \( m\angle L = 36^\circ \)
- \( m\angle M = 36^\circ \)
- \( m\angle N = 108^\circ \)
In summary:
- \( m\angle M = 36^\circ \)
- \( m\angle N = 108^\circ \)
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