Question

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

Explanation:

Step1: Recall the tangent-secant angle theorem

The measure of an angle formed by a tangent and a secant drawn from a point outside the circle is half the difference of the measures of the intercepted arcs. The formula is \( m\angle ABC=\frac{1}{2}(m\overset{\frown}{ADC}-m\overset{\frown}{AC}) \), where \( \overset{\frown}{ADC} \) is the major arc and \( \overset{\frown}{AC} \) is the minor arc.

Step2: Identify the measures of the arcs

From the diagram, the measure of the minor arc \( \overset{\frown}{AC} = 126^\circ \), and the measure of the major arc \( \overset{\frown}{ADC}=360^\circ - 126^\circ=234^\circ \) (since the total circumference of a circle corresponds to \( 360^\circ \)).

Step3: Apply the tangent - secant angle formula

Substitute the values of the arcs into the formula:
\( m\angle ABC=\frac{1}{2}(234^\circ - 126^\circ) \)
First, calculate the difference inside the parentheses: \( 234^\circ-126^\circ = 108^\circ \)
Then, take half of that difference: \( \frac{1}{2}\times108^\circ = 54^\circ \)

Answer:

\( 54^\circ \) (corresponding to the option "54°")