Question

use the figure below to find ( mangle abc ).

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. The inscribed angle is \(\angle ABC\) and the intercepted arc is \(\overset{\frown}{AC}\). The measure of the arc \(\overset{\frown}{AC}\) is \(94^{\circ}\). Also, we know that the sum of the arcs in a circle is \(360^{\circ}\), but since \(BC\) is a diameter, the arc \(\overset{\frown}{BAC}\) is \(180^{\circ}\). Let \(m\angle ABC = x\).

Step2: Apply the inscribed - angle formula

The formula for an inscribed angle \(\theta=\frac{1}{2}\text{(measure of intercepted arc)}\). The intercepted arc for \(\angle ABC\) is \(180^{\circ}- 94^{\circ}=86^{\circ}\). Using the formula \(m\angle ABC=\frac{1}{2}(180 - 94)\).

$$m\angle ABC=\frac{1}{2}\times86$$

$$m\angle ABC = 43^{\circ}$$

Answer:

\(43^{\circ}\)