Question

in the diagram of circle o, what is the measure of ∠abc?
○ 30°
○ 40°
○ 50°
○ 60°

Explanation:

Step1: Recall the theorem for angle formed by two tangents

The measure of an angle formed by two tangents drawn from an external point to a circle is half the difference of the measures of the intercepted arcs. The total measure of a circle is \(360^\circ\). We know one arc is \(210^\circ\) and the other (the minor arc \(AC\)) is \(150^\circ\). The formula for the angle \(\angle ABC\) (formed by two tangents from \(B\)) is \(\frac{1}{2}(\text{measure of major arc } - \text{measure of minor arc})\).

Step2: Calculate the difference of the arcs

First, find the difference between the major arc (\(210^\circ\)) and the minor arc (\(150^\circ\)): \(210^\circ - 150^\circ = 60^\circ\).

Step3: Apply the tangent - angle theorem

Then, take half of that difference: \(\frac{1}{2}\times60^\circ = 30^\circ\).

Answer:

\(30^\circ\) (corresponding to the option with \(30^\circ\))