Question

what is the measure of the highlighted arc? ab is a diameter.
enter deg after any value that is in degrees.

Explanation:

Step1: Use the inscribed - angle theorem

The inscribed - angle theorem states that an inscribed angle \(\angle ACB\) is half of the central angle that subtends the same arc. Also, since \(AB\) is a diameter, \(\angle ACB = 90^{\circ}\) (angle inscribed in a semi - circle). Given \(\angle CAB=40^{\circ}\), then \(\angle ABC = 180^{\circ}-\angle ACB-\angle CAB=180^{\circ}-90^{\circ}-40^{\circ} = 50^{\circ}\).

Step2: Use the arc - angle relationship

The measure of an inscribed angle is half the measure of its subtended arc. Let the measure of arc \(BC\) be \(x\). The inscribed angle \(\angle CAB\) subtends arc \(BC\). By the formula \(\angle CAB=\frac{1}{2}\text{arc}(BC)\), we know that \(\text{arc}(BC) = 2\angle CAB\). Let the measure of the highlighted arc (arc \(AC\)) be \(y\). The inscribed angle \(\angle ABC\) subtends arc \(AC\). Using the formula \(\angle ABC=\frac{1}{2}\text{arc}(AC)\) (inscribed - angle formula: \(\theta=\frac{1}{2}\alpha\), where \(\theta\) is the inscribed angle and \(\alpha\) is the subtended arc).
Since \(\angle ABC = 50^{\circ}\), then \(y = 2\angle ABC\).

Answer:

\(100\)