To find the angle at vertex \( N \) in right triangle \( LMN \), we can use the sine, cosine, or tangent functions. Since we know the lengths of the opposite side \( LM \) and the hypotenuse \( MN \), we can use the sine function.
The sine of angle \( N \) is given by the formula:
\[ \sin(N) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{LM}{MN} \]
Substituting the given values:
\[ \sin(N) = \frac{8}{10} = 0.8 \]
Now, we need to calculate the angle \( N \):
\[ N = \sin^{-1}(0.8) \]
Using a calculator:
\[ N \approx 53.13^\circ \]
Rounding to the nearest degree, the angle measure at vertex \( N \) is approximately \( 53^\circ \).
Thus, the answer is \( 53^\circ \).