In the following figure, \(m(\text{arc NS}) = 125^\circ\), \(m(\text{arc EF}) = 37^\circ\). Find \(m\angle NMS\).[Figure: circle with chords NS and EF intersecting at the point M inside the circle; N, F on the upper part of the circle, E and S on the…

Answer: \(m\angle NMS = 81^\circ\)

Assumption about the figure: the figure description says “chords NS and EF”, but if N, M, S lay on one chord, \(\angle NMS\) would be a straight angle. For \(\angle NMS\) to intercept arc NS (with its vertically opposite angle intercepting arc EF), the two chords meeting at M must be chord NE and chord SF (N and S are ends of different chords). The solution uses this configuration, which is the one in the original figure.

Theorem: the measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertically opposite angle.

\(\angle NMS\) intercepts arc NS, and its vertically opposite angle \(\angle EMF\) intercepts arc EF.

\(\therefore m\angle NMS = \dfrac{1}{2}\left[m(\text{arc NS}) + m(\text{arc EF})\right] = \dfrac{1}{2}(125^\circ + 37^\circ) = \dfrac{162^\circ}{2} = 81^\circ\).