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)

Responses

106°
106°

53°
53°

74°
74°

90°
90°

1 answer

In a circle, the inscribed angle is half the measure of the arc it intercepts. In this case, angle N intercepts the arc MN.

Given that angle L (which is 37°) is inscribed in the circle, we can determine that angle L represents an angle formed by the endpoints of the diameter LM and point N on the circumference. The arc measure of LM is equal to the whole circle (180°, since LM is the diameter), and N is the inscribed angle corresponding to arc MN.

To find the arc measure of MN, we can apply the property of inscribed angles:

\[ \text{Arc measure of } MN = 2 \times \text{Angle N} \]

Since angle L measures 37°, the inscribed angle N measures:

\[ \text{Angle N} = 90° - 37° = 53° \]

Now, we can find the arc measure of MN:

\[ \text{Arc measure of } MN = 2 \times 53° = 106° \]

Therefore, the arc measure of MN is 106°. The correct response is:

106°