Question

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

90°
90°

74°
74°

106°
106°

53°

Answer 1

In a circle, the measure of an inscribed angle is half the measure of the arc that it intercepts. In triangle LMN, angle N is inscribed, and it intercepts arc MN.

Given that angle L measures 37°, the opposite arc, which is arc MN, can be found using the following relationship:

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

Since angle L (37°) is at vertex L, we calculate the measure of angle N as follows:

\[
\text{The angles in triangle LMN sum up to 180°}
\]
\[
\text{If we denote angle M as } m \text{ and angle N as } n,
\]
\[
37° + m + n = 180°
\]
\[
m + n = 143°
\]

To find the inscribed angle N, let's use the relationship that angle N intercepts arc LM. Since LM has a measure equal to \( 180° \) (diameter):

\[
\text{Arc measure of } LM = 180° \implies m = 90° \text{ (by complementary angles)}
\]
\[
n = 143° - 90° = 53°
\]

Now, we want to find the arc MN which is intercepted by angle N:

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

Thus, the arc measure of MN is

\[
\boxed{106°}
\]