Question

in circle o, m\\(widehat{ac}=m\widehat{bc}=160\\). what is m\\(\angle aob\\)? (not drawn to scale)

Explanation:

Step1: Recall circle - arc relationship

The sum of the measures of the arcs of a circle is 360°.

Step2: Calculate the measure of arc \(AB\)

We know that \(m\widehat{AC}=m\widehat{BC} = 160^{\circ}\), and \(m\widehat{AC}+m\widehat{BC}+m\widehat{AB}=360^{\circ}\). Substitute the values: \(160^{\circ}+160^{\circ}+m\widehat{AB}=360^{\circ}\), then \(m\widehat{AB}=360^{\circ}-(160^{\circ} + 160^{\circ})=40^{\circ}\).

Step3: Recall central - angle and arc relationship

The measure of a central angle is equal to the measure of its intercepted arc. \(\angle AOB\) is a central angle that intercepts arc \(AB\). So \(m\angle AOB=m\widehat{AB}\).

Answer:

C. 40