Question

question 26 of 44
what is the measure of $\angle abc$, given that $\overline{ab}$ and $\overline{bc}$ are tangent to $\odot o$?

diagram: circle with center $o$, points $a$, $c$ on the circle, tangent lines $ab$ (at $a$) and $bc$ (at $c$), point $d$ on the circle, arc from $d$ to $a$ labeled $228^\circ$, arc from $c$ to $a$ (or $d$ to $c$) labeled $132^\circ$? \click here for long description\ below the diagram.

options:
a. $48^\circ$
b. $134^\circ$
c. $192^\circ$
d. $96^\circ$

Explanation:

Step1: Recall the property of tangents and the circle

The measure of an angle formed by two tangents drawn from an external point to a circle is equal to half the difference of the measures of the intercepted arcs. The total measure of a circle is \(360^\circ\). Given one arc \( \overset{\frown}{ADC} \) is \(228^\circ\), the other arc \( \overset{\frown}{AC} \) (the minor arc) can be found by subtracting from \(360^\circ\). So, \( m\overset{\frown}{AC}=360^\circ - 228^\circ=132^\circ \)? Wait, no, wait. Wait, the angle formed by two tangents \( \angle ABC \) is given by \( \angle ABC=\frac{1}{2}(\text{measure of the major arc } - \text{measure of the minor arc}) \). Wait, actually, the formula is \( \angle ABC=\frac{1}{2}(m\overset{\frown}{ADC}-m\overset{\frown}{AC}) \). Wait, let's correct. The measure of an angle formed by two tangents from an external point is \( \frac{1}{2} \) the difference of the measures of the intercepted arcs (the major arc and the minor arc). So first, find the measure of the minor arc \( AC \). The major arc \( ADC \) is \(228^\circ\), so the minor arc \( AC = 360^\circ - 228^\circ = 132^\circ\)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, the angle between two tangents: the formula is \( \angle B=\frac{1}{2}( \text{major arc } - \text{minor arc}) \). So major arc is \(228^\circ\), minor arc is \(360 - 228 = 132^\circ\)? Wait, no, that would make \( \angle B=\frac{1}{2}(228 - 132)=\frac{1}{2}(96)=48^\circ \)? Wait, no, that's not matching. Wait, maybe the diagram has the reflex angle? Wait, no, let's re-examine. Wait, the two tangents from B to the circle touch at A and C. So the angle at B, \( \angle ABC \), and the arcs: the measure of \( \angle ABC \) is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc AC and the minor arc AC. So if the major arc AC is \(228^\circ\), then the minor arc AC is \(360 - 228 = 132^\circ\)? Wait, no, that would mean the major arc is larger. Wait, \(228^\circ\) is larger than \(132^\circ\), so major arc is \(228^\circ\), minor is \(132^\circ\). Then \( \angle ABC=\frac{1}{2}(228 - 132)=\frac{1}{2}(96)=48^\circ \)? But that's option A. Wait, but let's check again. Wait, the formula for the angle between two tangents: \( \angle B = \frac{1}{2}( \text{measure of the major arc } - \text{measure of the minor arc}) \). So major arc \( AC \) is \(228^\circ\), minor arc \( AC \) is \(360 - 228 = 132^\circ\). Then \( \angle B = \frac{1}{2}(228 - 132) = \frac{1}{2}(96) = 48^\circ \). So the answer should be A. 48°.

Wait, but let's confirm the formula. The measure of an angle formed by two tangents drawn from an external point to a circle is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs are the two arcs that are cut off by the tangents and lie on the circle. So yes, the formula is \( \angle = \frac{1}{2}( \text{major arc} - \text{minor arc}) \). So with major arc \(228^\circ\) and minor arc \(132^\circ\), the angle is \( \frac{1}{2}(228 - 132) = 48^\circ \).

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

The total measure of a circle is \(360^\circ\). Given the major arc \( \overset{\frown}{ADC} = 228^\circ \), the minor arc \( \overset{\frown}{AC} = 360^\circ - 228^\circ = 132^\circ \).

Step2: Apply the tangent - angle formula

The measure of an angle formed by two tangents from an external point (here, \( B \)) to a circle is given by \( \angle ABC=\frac{1}{2}(\text{measure of major arc } - \text{measure of minor arc}) \). Substituting the values: \( \angle ABC=\frac{1}{2}(228^\circ - 132^\circ) \).

Step3: Calculate the angle

First, compute the difference: \(228^\circ - 132^\circ = 96^\circ\). Then, take half of that: \( \frac{96^\circ}{2}=48^\circ \).

Answer:

A. 48°