Since the triangles \( \Delta PQR \) and \( \Delta LMN \) are similar (denoted by \( \Delta PQR \sim \Delta LMN \)), their corresponding angles are equal.
We know:
- \( m \angle Q = 113^\circ \)
- \( m \angle R = 26^\circ \)
To find \( m \angle P \) in triangle \( PQR \), we use the fact that the sum of the angles in a triangle equals \( 180^\circ \):
\[ m \angle P + m \angle Q + m \angle R = 180^\circ \]
Substituting the known values:
\[ m \angle P + 113^\circ + 26^\circ = 180^\circ \]
Now, simplify:
\[ m \angle P + 139^\circ = 180^\circ \]
Subtract \( 139^\circ \) from both sides:
\[ m \angle P = 180^\circ - 139^\circ = 41^\circ \]
Now, since \( \Delta PQR \sim \Delta LMN \), the corresponding angle \( m \angle L \) will be equal to \( m \angle P \):
\[ m \angle L = 41^\circ \]
Next, to find \( m \angle N \), we use the angle sum property for triangle \( LMN \):
\[ m \angle L + m \angle M + m \angle N = 180^\circ \]
Since \( m \angle M \) corresponds to \( m \angle Q \):
\[ m \angle M = 113^\circ \]
Now substitute \( m \angle L \) and \( m \angle M \):
\[ 41^\circ + 113^\circ + m \angle N = 180^\circ \]
Combine the angles:
\[ 154^\circ + m \angle N = 180^\circ \]
Now, subtract \( 154^\circ \) from both sides:
\[ m \angle N = 180^\circ - 154^\circ = 26^\circ \]
Therefore, the measurement of angle \( N \) is:
\[ m \angle N = 26^\circ \]