Since triangles \( \triangle PQR \) and \( \triangle LMN \) are similar, their corresponding angles are equal.
Given that:
- \( m\angle Q = 113^\circ \)
- \( m\angle R = 26^\circ \)
We can find the measure of angle \( P \) in triangle \( PQR \) using 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 that we know the angles of triangle \( PQR \):
- \( m\angle P = 41^\circ \)
- \( m\angle Q = 113^\circ \)
- \( m\angle R = 26^\circ \)
Since \( \triangle PQR \sim \triangle LMN \), the corresponding angle \( N \) in triangle \( LMN \) corresponds to angle \( R \) in triangle \( PQR \), and we can find \( m\angle N \):
\[ m\angle N = m\angle R = 26^\circ \]
Thus, the measure of angle \( N \) is \( \boxed{26^\circ} \).
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