Question

in the figure below, o is the center of the circle. find each measurement below. a. m∠aco

Explanation:

Step1: Recall circle - related angle properties

In a circle with center \(O\), \(OA = OC\) since they are radii of the circle. So, \(\triangle AOC\) is isosceles.

Step2: Find \(\angle AOC\)

The inscribed - angle theorem states that the measure of an inscribed angle is half the measure of the central angle subtended by the same arc. Arc \(AC\) is subtended by \(\angle ABC\) (inscribed angle) and \(\angle AOC\) (central angle). Given \(\angle ABC = 58^{\circ}\), then \(\angle AOC=2\angle ABC = 116^{\circ}\).

Step3: Calculate \(\angle ACO\)

In \(\triangle AOC\), since \(OA = OC\), let \(\angle ACO=\angle CAO = x\). Using the angle - sum property of a triangle (\(\angle AOC+\angle ACO+\angle CAO = 180^{\circ}\)), we substitute \(\angle AOC = 116^{\circ}\) and \(\angle ACO=\angle CAO=x\) into the equation: \(116^{\circ}+x + x=180^{\circ}\), which simplifies to \(116^{\circ}+2x = 180^{\circ}\). Then \(2x=180^{\circ}-116^{\circ}=64^{\circ}\), and \(x = 32^{\circ}\).

Answer:

\(32\)