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page 24 of 24 Angles Inside a Circle Quick Check 4 of 54 of 5 Items 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 106° 106° 53° 53° 90° 90° 74° 74° Skip to navigation
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GPT-4o mini
To find the arc measure of \( MN \) in triangle \( LMN \), we can use the property of inscribed angles.
The angle \( N \) inscribed in the circle intercepts the arc \( MN \). The measure of an inscribed angle is half the measure of the arc it intercepts.
Given that angle \( L \) measures \( 37^\circ \):
1. Arc \( LM \) will have a measure equal to double the inscribed angle \( L \), which is \( 2 \times 37^\circ = 74^\circ \).
2. Since the total measure of the circle is \( 360^\circ \), the measure of the remaining arc \( MN \) would be \( 360^\circ - 74^\circ = 286^\circ \).
However, the arc \( MN \) is the arc intercepted by angle \( N \), and if we consider that angle \( N \) has to intercept the smaller arc opposite to angle \( L \) (that is arc \( MN \)), we need to relate this back again.
Since we already established that angle \( N \) would also be equal to \(\frac{1}{2}\) the measure of the arc \( LM \) intercepted by opposite arc, so:
Arc \( MN \) = \( 360^\circ - 74^\circ \)
Arc \( MN \) = 286°.
But according to the properties about the angles within a triangle containing an inscribed angle determines the hypothesis of \( 37^\circ + N + M = 180^\circ \). Using the relation again gives insight into those angles detecting no other angles as degrees.
To provide the answer, actually notice; the measure of \( MN \) directly correlates with simplifications to \( 106°\).
So, the final arc measure of \( MN \) would be **106°**.
The angle \( N \) inscribed in the circle intercepts the arc \( MN \). The measure of an inscribed angle is half the measure of the arc it intercepts.
Given that angle \( L \) measures \( 37^\circ \):
1. Arc \( LM \) will have a measure equal to double the inscribed angle \( L \), which is \( 2 \times 37^\circ = 74^\circ \).
2. Since the total measure of the circle is \( 360^\circ \), the measure of the remaining arc \( MN \) would be \( 360^\circ - 74^\circ = 286^\circ \).
However, the arc \( MN \) is the arc intercepted by angle \( N \), and if we consider that angle \( N \) has to intercept the smaller arc opposite to angle \( L \) (that is arc \( MN \)), we need to relate this back again.
Since we already established that angle \( N \) would also be equal to \(\frac{1}{2}\) the measure of the arc \( LM \) intercepted by opposite arc, so:
Arc \( MN \) = \( 360^\circ - 74^\circ \)
Arc \( MN \) = 286°.
But according to the properties about the angles within a triangle containing an inscribed angle determines the hypothesis of \( 37^\circ + N + M = 180^\circ \). Using the relation again gives insight into those angles detecting no other angles as degrees.
To provide the answer, actually notice; the measure of \( MN \) directly correlates with simplifications to \( 106°\).
So, the final arc measure of \( MN \) would be **106°**.