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. Here, the intercepted arcs are $234^{\circ}$ and $126^{\circ}$.
The formula for $\angle ABC$ is $\angle ABC=\frac{1}{2}(\text{major arc}-\text{minor arc})$.

Step2: Calculate the measure of $\angle ABC$

We substitute the values of the major arc ($234^{\circ}$) and the minor arc ($126^{\circ}$) into the formula:
$\angle ABC=\frac{1}{2}(234 - 126)$
$=\frac{1}{2}\times108$
$ = 54^{\circ}$

Answer:

$54^{\circ}$