Question

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

Explanation:

Step1: Recall the Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Step2: Identify the relationship

Given that \(\angle BAC = 61^{\circ}\), and \(\angle BAC\) is an inscribed angle. Let the measure of arc \(BC\) be \(x\). Then \(\angle BAC=\frac{1}{2}x\). So \(x = 2\times61^{\circ}=122^{\circ}\).

Step3: Use the property of a semicircle

Since \(AC\) is a diameter, the measure of arc \(AC\) is \(180^{\circ}\). Let the measure of the highlighted arc \(AB\) be \(y\). We know that \(y+122^{\circ}=180^{\circ}\) (because arc \(AB\) + arc \(BC\)= arc \(AC\)). Then \(y = 180^{\circ}-122^{\circ}=58^{\circ}\).

Answer:

\(58\mathrm{deg}\)