To find the arc measure of \( MN \) when triangle \( LMN \) is inscribed in a circle with diameter \( LM \) and the angle at vertex \( L \) measures \( 37^\circ \), we can use the properties of inscribed angles.
The inscribed angle \( \angle L \) intercepts the arc \( MN \). According to the properties of inscribed angles, the measure of an inscribed angle is half the measure of the arc it intercepts.
Let \( m \) be the measure of arc \( MN \). According to the relationship:
\[ \angle L = \frac{1}{2} m \]
Given that \( \angle L = 37^\circ \), we can set up the equation:
\[ 37^\circ = \frac{1}{2} m \]
To find \( m \), we multiply both sides by 2:
\[ m = 2 \times 37^\circ = 74^\circ \]
Thus, the measure of the arc \( MN \) is \( 74^\circ \).
There may be a misunderstanding with the response of \( 106^\circ \), as \( 74^\circ \) is the correct measure for arc \( MN \).
If you're looking for the measure of the remaining arc (which may be referred to if a larger angle is being considered), the total circumference of a circle is \( 360^\circ \). Therefore, if the full circle is \( 360^\circ \):
\[ \text{Measure of the remaining arc} = 360^\circ - 74^\circ = 286^\circ \]
But as per the original question focusing on arc \( MN \), the arc measure of \( MN \) is \( 74^\circ \).
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