To find the angle measure at vertex N in triangle LMN, we can use the cosine rule or trigonometric ratios. However, since we already have the lengths of the sides, we can directly use the sine or cosine functions.
In right triangle LMN, with \( \angle L \) being the right angle, we know:
- \( LM = 8 \) units (adjacent side to angle N),
- \( NL = 6 \) units (opposite side to angle N),
- \( MN = 10 \) units (hypotenuse).
We can use the sine function to find the angle \( N \):
\[ \sin(N) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{NL}{MN} = \frac{6}{10} = 0.6 \]
Now, we calculate angle \( N \) using the inverse sine function:
\[ N = \arcsin(0.6) \]
Calculating using a calculator:
\[ N \approx 36.87^\circ \]
Thus, the approximate angle measure at vertex N is \( \boxed{36.87} \) degrees.