Question

what is the measure of the highlighted arc? \\( \overline { a c } \\) is a diameter.
enter deg after any value that is in degrees.

Explanation:

Step1: Recall the relationship between central angle and arc measure

The measure of an arc is equal to the measure of its central angle.

Step2: Calculate the measure of the arc

Since \(AC\) is a diameter, the total measure of the circle is \(360^{\circ}\). The un - highlighted part of the arc (the minor arc \(AB\)) has a central angle of \(40^{\circ}\). The highlighted arc is a major arc. The measure of a major arc \(=\) \(360^{\circ}-\) measure of the minor arc. But another way: If we consider the fact that the measure of an arc is equal to the measure of its central angle. The central angle for the highlighted arc (major arc \(AB\)): The sum of angles around a point is \(360^{\circ}\). The central angle for the major arc \(AB\) can also be calculated as \(180^{\circ}+(180^{\circ}- 40^{\circ}-40^{\circ})\) (using the property of inscribed angles in a semicircle, but more simply, since the central angle for the arc \(AB\) (major) is \(180^{\circ}+40^{\circ}\) (because the angle subtended by an arc at the center is related to the inscribed angle. In a circle, if we know that the angle at the circumference is half of the angle at the center subtended by the same arc. But for a diameter \(AC\), the central angle for the semicircle is \(180^{\circ}\). The measure of the arc \(AB\) (highlighted) is \(180^{\circ}+40^{\circ}\)

$$180^{\circ}+40^{\circ}=220^{\circ}$$

Answer:

\(220^{\circ}\)