Question

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

Explanation:

Step1: Recall external angle theorem

The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs.

Step2: Identify intercepted arcs

The intercepted arcs are the major arc $\overset{\frown}{AC}$ ($234^\circ$) and the minor arc $\overset{\frown}{AC}$ ($126^\circ$).

Step3: Calculate the angle

$\angle ABC = \frac{1}{2} \times (\text{measure of major arc } \overset{\frown}{AC} - \text{measure of minor arc } \overset{\frown}{AC})$
$\angle ABC = \frac{1}{2} \times (234^\circ - 126^\circ)$
$\angle ABC = \frac{1}{2} \times 108^\circ$

Answer:

$54^\circ$ (Option B. $54^\circ$)