Since triangles \( \triangle PQR \) and \( \triangle LMN \) are similar (denoted by \( \triangle PQR \sim \triangle LMN \)), the corresponding angles are equal.
In triangle \( PQR \):
- \( m \angle Q = 113^\circ \)
- \( m \angle R = 26^\circ \)
To find the measure of angle \( P \), we can use the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[ m \angle P + m \angle Q + m \angle R = 180^\circ \] \[ m \angle P + 113^\circ + 26^\circ = 180^\circ \] \[ m \angle P + 139^\circ = 180^\circ \] \[ m \angle P = 180^\circ - 139^\circ = 41^\circ \]
Now, in triangle \( LMN \), the corresponding angles must match:
- \( m \angle L = m \angle P = 41^\circ \)
- \( m \angle M = m \angle Q = 113^\circ \)
- \( m \angle N = m \angle R = 26^\circ \)
Therefore, the measure of angle \( N \) is:
\[ m \angle N = 26^\circ \]
So, \( m \angle N = 26^\circ \).