Since triangles \( \triangle PQR \) and \( \triangle LMN \) are similar (denoted as \( \triangle PQR \sim \triangle LMN \)), their corresponding angles are equal.
Given:
- \( m\angle Q = 113^\circ \)
- \( m\angle R = 26^\circ \)
To find \( m\angle P \) in triangle \( PQR \), we can use the fact that the sum of the angles in a triangle is \( 180^\circ \).
\[ m\angle P + m\angle Q + m\angle R = 180^\circ \] \[ m\angle P + 113^\circ + 26^\circ = 180^\circ \] \[ m\angle P + 139^\circ = 180^\circ \] \[ m\angle P = 180^\circ - 139^\circ = 41^\circ \]
Since the triangles are similar, \( m\angle L = m\angle P \).
Now, to find \( m\angle N \) in triangle \( LMN \), we can again use the fact that the sum of angles in the triangle must be \( 180^\circ \):
\[ m\angle L + m\angle M + m\angle N = 180^\circ \]
Since we don't know \( m\angle M \) yet, we also apply the property of similarity. Since \( \triangle PQR \sim \triangle LMN \), we have
\[ m\angle M = m\angle Q = 113^\circ \]
Now substituting, we get
\[ 41^\circ + 113^\circ + m\angle N = 180^\circ \] \[ 154^\circ + m\angle N = 180^\circ \] \[ m\angle N = 180^\circ - 154^\circ = 26^\circ \]
Thus, the measure of angle \( N \) is
\[ \boxed{26^\circ} \]