Question

in the figure below, o is the center of the circle. find each measurement below.
a. m∠aco
b. m\\(overset{\frown}{ac})
c. m\\(overset{\frown}{cb})

Explanation:

Step1: Recall circle - related angle rules

The measure of an inscribed - angle is half the measure of its intercepted arc. The sum of the measures of arcs in a circle is 360°.

Step2: Find the measure of arc \(AC\)

We know that the central angle corresponding to arc \(AC\) is \(2\times\angle ABC\). Given \(\angle ABC = 58^{\circ}\), then \(m\widehat{AC}=2\times58^{\circ}=116^{\circ}\) (the central - angle is twice the inscribed - angle subtended by the same arc).

Step3: Find the measure of arc \(CB\)

We know that \(m\widehat{AC}+m\widehat{CB}+m\widehat{BA}=360^{\circ}\). Since \(m\widehat{AC} = 116^{\circ}\) and the diameter \(AB\) subtends an arc \(m\widehat{AB}=180^{\circ}\), then \(m\widehat{CB}=360^{\circ}-m\widehat{AC}-m\widehat{AB}\). Substituting the values, we get \(m\widehat{CB}=360^{\circ}-116^{\circ}-180^{\circ}=64^{\circ}\).

Answer:

c. \(m\widehat{CB}=64^{\circ}\)