Question

if a central angle nam, measures 59°, what is the measure of circumscribed angle nbm?
○ 59°
○ 239°
○ 121°
○ 301°

Explanation:

Step1: Recall the property of circumscribed angle and central angle

A circumscribed angle (tangent angle) and the central angle subtended by the same arc are supplementary if we consider the related arcs, but more accurately, for a circumscribed angle formed by two tangents, the measure of the circumscribed angle is equal to \(180^{\circ}-\) the measure of the central angle subtended by the intercepted arc. Wait, actually, the formula for the measure of a circumscribed angle (formed by two tangents) with respect to a central angle \(\theta\) subtended by the same arc is \(180^{\circ}-\theta\) when the central angle is \(\theta\) (since the sum of the central angle and the circumscribed angle related to the same arc and tangents is \(180^{\circ}\) in the case of two tangents forming the circumscribed angle and the central angle between the radii to the points of tangency). Wait, let's correct: The measure of a circumscribed angle (angle formed by two tangents) is equal to \(180^{\circ}-\) the measure of the central angle that intercepts the same arc. So if central angle \(NAM = 59^{\circ}\), then the circumscribed angle \(NBM\) (assuming \(B\) is the point of tangency, and \(N\) and \(M\) are the points on the circle, with \(A\) the center) is \(180 - 59\).

Step2: Calculate the measure of the circumscribed angle

\[180^{\circ}- 59^{\circ}=121^{\circ}\]

Answer:

\(121^{\circ}\) (corresponding to the option with \(121^{\circ}\))