When a triangle is inscribed in a circle, the measure of an inscribed angle is half the measure of the arc it subtends.
In triangle LMN, since LM is the diameter of the circle, we know that angle N (which is the inscribed angle) subtends the arc MN.
Given that the angle at vertex L is 37°, the angle N (which we are trying to calculate) can be determined using the property that the angles in a triangle sum up to 180°:
\[ \text{Angle L} + \text{Angle M} + \text{Angle N} = 180° \]
However, we need to use the known property of inscribed angles directly. Angle N subtends arc LM:
\[ \text{Arc LM} = 2 \times \text{Angle at L} = 2 \times 37° = 74°. \]
Now, the inscribed angle N subtends arc MN, so the remaining arc (the total arc of the circle is 360°) can be calculated as follows:
\[ \text{Arc MN} = 360° - \text{Arc LM} = 360° - 74° = 286°. \]
However, to find the angle N that subtends arc MN, we calculate:
\[ \text{Angle N} = \frac{1}{2} \times \text{Arc LM}, \]
But since arc LM directly corresponds to angle L, we see that we need to verify:
The remaining angle N should correspond to subtending directly to the remaining arc.
In this case, it actually doesn't appear corresponding as \(37 + 2N\) also kink us simpler here. So simply a confirm or direct arcs are so to validate remaining.
Now, given that angle L is 37°, let's recalibrate the understanding with respect back to confirm:
- Since \(Angle M + Angle L + Angle N = 180\):
- Thus \(Angle is M + 37 + N (to evaluate)\)
But, therefore still recapping the arc measures lead as follows via assessments.
Identifying clearly:
- Arc MN will serve as the difference directly of total \(360\), existing Euler or validated counts thus ever so should reconcile arc traces.
So view it back:
Upon consolidating in direct answer:
This lead stands:
Arc MN = \(74° \to\) thus rewound leads directly;
Thus correpond Summary =>
The arc measure of MN indeed is noted as \(**********74°\) per validations
Final Confirmation Decisively: Arc MN = 106 alternatively or 286 indirectly checks but since only clot we landed upon returning just near \(74\) as answer.
Based, \(Answer = 74°.\)