Question

in the figure below, o is the center of the circle. find each measurement below.
a. m∠aco
b. mac
c. mcb
d. mab
e. mcba

Explanation:

Step1: Recall inscribed - angle and central - angle relationships

The measure of an inscribed angle is half the measure of the central angle subtended by the same arc. Also, the sum of the measures of the arcs of a circle is 360°.

Step2: Find \(m\angle ACO\)

Given directly as \(32^{\circ}\).

Step3: Find \(m\widehat{AC}\)

The central angle corresponding to arc \(\widehat{AC}\) has a measure equal to the sum of the angles in \(\triangle AOC\) at the center. Since \(OA = OC\) (radii of the circle), \(\angle OAC=\angle ACO = 32^{\circ}\), and the central - angle \(m\angle AOC=180-(32 + 32)=116^{\circ}\). So \(m\widehat{AC}=116^{\circ}\).

Step4: Find \(m\widehat{CB}\)

The inscribed angle \(\angle CAB\) subtends arc \(\widehat{CB}\). Given \(\angle CAB = 32^{\circ}\), then \(m\widehat{CB}=2\times\angle CAB=64^{\circ}\).

Step5: Find \(m\widehat{AB}\)

Given as \(180^{\circ}\).

Step6: Find \(m\widehat{CBA}\)

The sum of the arcs of a circle is \(360^{\circ}\). We know \(m\widehat{AC}=116^{\circ}\), \(m\widehat{CB}=64^{\circ}\), and \(m\widehat{AB}=180^{\circ}\). The arc \(\widehat{CBA}=\widehat{CB}+\widehat{BA}\), so \(m\widehat{CBA}=64 + 180=244^{\circ}\).

Answer:

\(244\)