Question

in the diagram below, \\(\overleftrightarrow{ab}\\) and \\(\overleftrightarrow{bc}\\) are tangent to circle o. what is the measure of \\(\angle b\\)?\
\\(\circ\\) 180°\
\\(\circ\\) 108°\
\\(\circ\\) 54°\
\\(\circ\\) 126°

Explanation:

Step1: Find the measure of the minor arc \( \overset{\frown}{AC} \)

The total circumference of a circle corresponds to \( 360^\circ \). Given the major arc \( \overset{\frown}{ADC} \) is \( 234^\circ \), the minor arc \( \overset{\frown}{AC} \) is \( 360^\circ - 234^\circ = 126^\circ \).

Step2: Use the tangent - angle formula

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. For angle \( \angle B \), the formula is \( \angle B=\frac{1}{2}(\text{measure of major arc }-\text{measure of minor arc}) \).
Substituting the values, we have \( \angle B=\frac{1}{2}(234^\circ - 126^\circ) \).

Step3: Calculate the measure of \( \angle B \)

First, calculate the difference inside the parentheses: \( 234^\circ - 126^\circ = 108^\circ \). Then, take half of that: \( \frac{1}{2}\times108^\circ = 54^\circ \). Wait, no, wait. Wait, the formula is \( \angle B = 180^\circ-\frac{1}{2}\times\text{major arc} \)? No, correct formula: The angle between two tangents from an external point is equal to \( 180^\circ-\frac{1}{2}\times \text{measure of the major arc} \)? Wait, no, let's re - derive.

Let \( O \) be the center of the circle. Let the points of tangency be \( A \) and \( C \). Then \( OA\perp BA \) and \( OC\perp BC \) (tangent to a circle is perpendicular to the radius at the point of tangency). So quadrilateral \( OABC \) has two right angles at \( A \) and \( C \). The sum of interior angles of a quadrilateral is \( 360^\circ \). Let \( \angle B=x \), \( \angle AOC = y \) (measure of arc \( AC \)). Then \( 90^\circ+90^\circ + x + y=360^\circ \), so \( x + y = 180^\circ \), so \( x=180^\circ - y \). But \( y \) is the measure of arc \( AC \). Wait, the major arc \( ADC \) is \( 234^\circ \), so the minor arc \( AC \) is \( 360 - 234=126^\circ \), so \( y = 126^\circ \)? No, wait, no. Wait, the angle between two tangents is equal to \( 180^\circ-\frac{1}{2}\times \text{measure of the major arc} \)? Wait, no, let's use the correct formula: The measure of an angle formed by two tangents to a circle is \( \frac{1}{2}(\text{measure of the major intercepted arc}-\text{measure of the minor intercepted arc}) \).

Major arc \( ADC = 234^\circ \), minor arc \( AC=360 - 234 = 126^\circ \). Then \( \angle B=\frac{1}{2}(234 - 126)=\frac{1}{2}(108) = 54^\circ \)? Wait, but that contradicts the initial thought. Wait, no, let's check again.

Wait, the formula for the angle between two tangents: If two tangents are drawn from an external point \( B \) to a circle with center \( O \), touching the circle at \( A \) and \( C \), then \( \angle OAB=\angle OCB = 90^\circ \), and \( OA = OC \) (radii). So quadrilateral \( OABC \) has \( \angle OAB+\angle OCB+\angle ABC+\angle AOC = 360^\circ \). So \( 90 + 90+\angle B+\angle AOC=360 \), so \( \angle B+\angle AOC = 180 \), so \( \angle B=180-\angle AOC \). But \( \angle AOC \) is the central angle corresponding to arc \( AC \). The major arc \( ADC \) is \( 234^\circ \), so the central angle for major arc \( ADC \) is \( 234^\circ \), and the central angle for minor arc \( AC \) is \( 360 - 234 = 126^\circ \), so \( \angle AOC = 126^\circ \)? No, wait, no. The central angle for arc \( AC \) (minor arc) is \( 126^\circ \), so \( \angle B=180 - 126=54^\circ \)? Wait, but let's check the answer options. 54 is an option. Wait, but maybe I made a mistake. Wait, the angle between two tangents is equal to \( 180^\circ-\frac{1}{2}\times \text{measure of the major arc} \)? Wait, no, let's use the formula: \( \angle B=\frac{1}{2}(\text{major arc}-\text{minor arc}) \). Major arc is \( 234^\circ \), minor arc is \( 126^\circ \), so \( \frac{234 - 126}{2}=\frac{108}{2}=54^\circ \). Yes, that's correct.

Answer:

\( 54^\circ \) (corresponding to the option with \( 54^\circ \))