In a circle, an inscribed angle is half the measure of the arc that it intercepts. In triangle LMN, the inscribed angle at vertex N (which is the angle °) intercepts arc MN.
We are given:
- The measure of angle L is 37°.
Since LM is the diameter, according to the inscribed angle theorem, angle N, which is opposite to diameter LM, measures 90° (as it subtends the semicircle).
Now, we can set up the relationship:
- The inscribed angle N intercepts arc MN, and we need to find its measure.
The measure of angle N can be calculated as:
- \( m \angle N = m \text{arc } MN / 2 \)
Since angle N is a right angle (90°):
- \( 90° = m \text{arc } MN / 2 \)
To find the measure of arc MN, we can multiply both sides by 2:
- \( m \text{arc } MN = 90° \times 2 = 180° \)
Next, we know that angles around point L must also take into consideration angle L which measures 37° and angle M as part of the triangle summing up to:
- 180° (the total angle sum in triangle LMN).
Thus,
- \( m \angle L + m \angle M + m \angle N = 180° \)
- \( 37° + m \angle M + 90° = 180° \)
Solving for \( m \angle M \):
- \( m \angle M = 180° - 127° = 53° \)
Now the arc measure for MN corresponds to the opposite angle measure from arc LM which equals twice \( m \angle M\):
- \( m \text{arc } MN = 2 \times 53° = 106°\)
Thus the correct answer:
106°