Question

what is the measure of ∠abc? 62° 88° 92° 176° 31°

Explanation:

Step1: Recall the inscribed angle theorem

The inscribed angle theorem states that an angle subtended by an arc at the center is twice the angle subtended at the circumference. Also, the measure of an inscribed angle is half the measure of its subtended arc. But here, angle at \( B \) is an inscribed angle? Wait, no, actually, the arc \( AC \) is given as \( 62^\circ \)? Wait, no, the arc \( AC \) is \( 62^\circ \), and we need to find the inscribed angle \( \angle ABC \). Wait, no, actually, the central angle for arc \( AC \) would be \( 62^\circ \), but the inscribed angle over arc \( AC \) would be half? Wait, no, maybe I got it wrong. Wait, the angle at \( B \): in a circle, the sum of the measures of an inscribed angle and the arc opposite? Wait, no, let's think again. The total around a point is \( 360^\circ \), but maybe the arc \( AC \) is \( 62^\circ \), and the angle at \( B \) is an inscribed angle. Wait, no, the inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. Wait, but if the arc \( AC \) is \( 62^\circ \), then the inscribed angle over arc \( AC \) would be \( \frac{62^\circ}{2}=31^\circ \)? Wait, but that's one of the options (31°). Wait, let's check the options. The options are 88°, 92°, 176°, 31°. So maybe the arc \( AC \) is \( 62^\circ \), and the angle \( \angle ABC \) is an inscribed angle intercepting arc \( AC \), so \( \angle ABC=\frac{1}{2}\times62^\circ = 31^\circ \). Wait, that makes sense. So step by step:

Step1: Identify the intercepted arc

The angle \( \angle ABC \) is an inscribed angle that intercepts arc \( AC \). The measure of arc \( AC \) is given as \( 62^\circ \).

Step2: Apply the inscribed angle theorem

The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. So, \( \angle ABC=\frac{1}{2}\times\text{measure of arc } AC \).

Substitute the measure of arc \( AC = 62^\circ \):

\( \angle ABC=\frac{1}{2}\times62^\circ = 31^\circ \)

Answer:

\( 31^\circ \) (so the option with \( 31^\circ \))