Given that triangles \( \triangle PQR \sim \triangle LMN \), it means that the corresponding angles of the triangles are equal.
We know the following measures:
- \( m\angle Q = 113^\circ \)
- \( m\angle R = 26^\circ \)
To find \( m\angle N \), we can first find the measure of angle \( P \) in triangle \( PQR \). The sum of angles in any triangle is \( 180^\circ \). Therefore:
\[ m\angle P + m\angle Q + m\angle R = 180^\circ \]
Substituting the known angle measures:
\[ m\angle P + 113^\circ + 26^\circ = 180^\circ \]
Adding the given angles:
\[ m\angle P + 139^\circ = 180^\circ \]
Now, solving for \( m\angle P \):
\[ m\angle P = 180^\circ - 139^\circ = 41^\circ \]
Since \( \triangle PQR \sim \triangle LMN \), we have:
- \( m\angle P = m\angle L \)
- \( m\angle Q = m\angle M \)
- \( m\angle R = m\angle N \)
Thus,
\[ m\angle L = 41^\circ, \quad m\angle M = 113^\circ, \quad m\angle N = 26^\circ \]
Now, to find \( m\angle N \):
Since \( m\angle R = m\angle N \), we know:
\[ m\angle N = m\angle R = 26^\circ \]
Therefore, the measure of angle \( N \) is:
\[ \boxed{41^\circ} \]