Question

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

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant is half the difference of the measures of the intercepted arcs.
The two intercepted arcs are $234^{\circ}$ and $126^{\circ}$.

Step2: Calculate the measure of $\angle ABC$

The formula for the measure of $\angle ABC$ is $\frac{1}{2}(m\overset{\frown}{major}-m\overset{\frown}{minor})$. Here, $m\overset{\frown}{major} = 234^{\circ}$ and $m\overset{\frown}{minor}=126^{\circ}$. So, $\angle ABC=\frac{1}{2}(234 - 126)$.
First, calculate $234-126 = 108$. Then, $\frac{1}{2}\times108 = 54^{\circ}$.

Answer:

$54^{\circ}$ (corresponding to the second option in the multiple - choice list)