Question

what is the measure of \\(\overarc{abc}\\)? (blue arc)
options: \\(310^\circ\\), \\(100^\circ\\), \\(260^\circ\\), \\(130^\circ\\)

Explanation:

Step1: Identify the inscribed angle

The angle at \( B \) (\( \angle ABC \)) is an inscribed angle. The measure of an inscribed angle is half the measure of its intercepted arc. Here, \( \angle ABC = 50^\circ \), so the intercepted arc \( \overset{\frown}{AC} \) has a measure of \( 2\times50^\circ = 100^\circ \).

Step2: Find the measure of the major arc \( \overset{\frown}{ABC} \)

The total measure of a circle is \( 360^\circ \). To find the measure of the major arc \( \overset{\frown}{ABC} \), we subtract the measure of the minor arc \( \overset{\frown}{AC} \) from \( 360^\circ \). So, \( 360^\circ - 100^\circ = 260^\circ \).

Answer:

\( 260^\circ \) (corresponding to the option with \( 260^\circ \))