Question

question 3 of 5
type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar.
segment ab is tangent to the circle at point a. what is the measure of ∠abc?
m∠abc = \\(\square\\) °

Explanation:

Step1: Find arc AC measure

The total circle is $360^\circ$. Calculate arc $AC$:
$m\overset{\frown}{AC} = 360^\circ - 100^\circ - 175^\circ - 85^\circ = 0^\circ$
*Correction: Identify the external angle rule. The external angle formed by a tangent and secant is half the difference of the intercepted arcs. The intercepted arcs are $\overset{\frown}{AC}$ and $\overset{\frown}{AD C}$. First calculate $\overset{\frown}{AC}$ correctly:
$m\overset{\frown}{AC} = 360^\circ - 175^\circ - 85^\circ - 100^\circ$ is incorrect. Instead, the arc intercepted outside is $\overset{\frown}{AC}$, the arc intercepted inside is $\overset{\frown}{ADC} = 100^\circ + 85^\circ + 175^\circ$? No, correct: $\overset{\frown}{ADC} = 175^\circ + 85^\circ + 100^\circ$ is wrong. The arc opposite to the external angle: the tangent touches at A, secant goes through C, so the intercepted arcs are $\overset{\frown}{AC}$ and the major arc $\overset{\frown}{ADC}$.
$m\overset{\frown}{AC} = 360^\circ - (175^\circ + 85^\circ + 100^\circ)$ is wrong, actually:
$175+85+100=360$, so $\overset{\frown}{AC}$ is not present. Wait, the arcs are $\overset{\frown}{AD}=175^\circ$, $\overset{\frown}{DC}=85^\circ$, $\overset{\frown}{CA}=100^\circ$? No, the labels: A to arc 175 to D, D to 85 to C, C to 100 to A. So total $175+85+100=360$. So the secant is BC, intersecting circle at C and D, tangent at A. The external angle $\angle ABC$ intercepts arc $\overset{\frown}{AC}$ and arc $\overset{\frown}{AD}$.
Wait, correct theorem: The measure of an angle formed by a tangent and a secant drawn from an external point is half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc and minor arc that lie between the two sides of the angle.
So $\angle ABC$ intercepts $\overset{\frown}{AC}$ (minor) and $\overset{\frown}{ADC}$ (major). $\overset{\frown}{AC}=100^\circ$, $\overset{\frown}{ADC}=360^\circ-100^\circ=260^\circ$. No, that can't be. Wait no: the tangent is AB, secant is BD (passing through C). So the two intercepted arcs are $\overset{\frown}{AC}$ and $\overset{\frown}{AD}$.
$\overset{\frown}{AD}=175^\circ+85^\circ=260^\circ$, $\overset{\frown}{AC}=100^\circ$.

Step2: Apply tangent-secant angle theorem

The formula is $m\angle ABC = \frac{1}{2}(m\overset{\frown}{AD} - m\overset{\frown}{AC})$
Substitute values:
$m\angle ABC = \frac{1}{2}(260^\circ - 100^\circ) = \frac{1}{2}(160^\circ) = 80^\circ$
*Correction: Wait, no, the correct intercepted arcs are the ones not containing the angle. The external angle is half the difference of the intercepted arcs: the larger arc minus the smaller arc that are cut off by the secant and tangent.
Wait, correct: $\overset{\frown}{ADC} = 175+85=260$, $\overset{\frown}{AC}=100$. So $\frac{1}{2}(260-100)=80$.

Answer:

80