Question

in the diagram, point o is the center of the circle. if ( mangle abc = 70^{circ} ), what is ( mangle aoc )?
a. ( 100^{circ} )
b. ( 110^{circ} )
c. ( 140^{circ} )
d. ( 130^{circ} )

Explanation:

Step1: Recall the relationship between central angle and inscribed angle

The measure of a central angle is twice the measure of an inscribed angle that subtends the same arc.

Step2: Identify the inscribed angle and the central angle

Here, \(\angle ABC\) is an inscribed angle and \(\angle AOC\) is the central angle subtending the same arc \(AC\).

Step3: Calculate the measure of \(\angle AOC\)

Since \(m\angle ABC = 70^{\circ}\), then \(m\angle AOC=2\times m\angle ABC\)
\(m\angle AOC = 2\times70^{\circ}=140^{\circ}\)

Answer:

C. \(140^{\circ}\)