Triangle LMN is inscribed inside a circle with diameter LM; the inscribed angle is N. The angle measure at the vertex L is 37°. Using what you understand about inscribed angles, find the arc measure of MN.(1 point)

Question

Responses

74°
74°

90°
90°

53°
53°

106°

Answer 1

In a circle, an inscribed angle (such as angle N in triangle LMN) is half the measure of the intercepted arc (arc MN, in this case).

Given that the angle at vertex L (∠L) measures 37°, we can determine the arc measures:

1. The inscribed angle ∠N intercepts arc MN.
2. The measure of the inscribed angle is equal to half the measure of the intercepted arc:

\[ m\angle N = \frac{1}{2} m\text{(arc MN)} \]

To find the measure of arc MN, we first note that since ∠L is 37°, the other angles in triangle LMN must sum to 180°. However, we can also find the measure of arc MN directly in relation to angle N.

Using the property of the inscribed angle:

• If ∠L measures 37°, this will form an exterior angle with angle N, hence angle N will also be equal to 37° (because angles on a straight line sum to 180°).

Thus: \[ m\angle N = 37° \Rightarrow m\text{(arc MN)} = 2 \times m\angle N = 2 \times 37° = 74° \]

Therefore, the measure of arc MN is \( 74° \).

The correct answer is: 74°