Question

in the diagram of circle o, what is the measure of ∠abc? 54° 108° 120° 27°

Explanation:

Step1: Recall inscribed - angle formula

The measure of an inscribed angle is half the measure of its intercepted arc. Here, $\angle ABC$ is an inscribed angle and the intercepted arc is $\overset{\frown}{AC}$. The measure of the major arc $\overset{\frown}{AC}=234^{\circ}$ and the measure of the minor arc $\overset{\frown}{AC} = 126^{\circ}$. The inscribed - angle formula for the angle formed by two tangents or a tangent and a secant is $\angle ABC=\frac{1}{2}(\text{major arc}-\text{minor arc})$.

Step2: Calculate the measure of $\angle ABC$

We know that the major arc $\overset{\frown}{AC} = 234^{\circ}$ and the minor arc $\overset{\frown}{AC}=126^{\circ}$. Then $\angle ABC=\frac{1}{2}(234 - 126)$.
First, calculate $234-126 = 108$. Then $\frac{1}{2}\times108 = 54$.

Answer:

$54^{\circ}$