In triangle \( \Delta LMN \), since it is isosceles with \( \angle M \cong \angle L \), we have:
- \( m\angle L = m\angle M = 36^\circ \)
To find the measure of angle \( N \), we can use the triangle sum property, which states that the sum of the angles in a triangle is always \( 180^\circ \).
So, we can write the equation:
\[ m\angle L + m\angle M + m\angle N = 180^\circ \]
Substituting the known values:
\[ 36^\circ + 36^\circ + m\angle N = 180^\circ \]
This simplifies to:
\[ 72^\circ + m\angle N = 180^\circ \]
Now, subtract \( 72^\circ \) from both sides:
\[ m\angle N = 180^\circ - 72^\circ = 108^\circ \]
Thus, we find the measures of the angles:
- \( m\angle M = 36^\circ \)
- \( m\angle N = 108^\circ \)
In conclusion:
- \( m\angle M = 36^\circ \)
- \( m\angle N = 108^\circ \)
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