Question

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 = 45 × °

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant is half the difference of the measures of the intercepted arcs. Let the major arc be $m\overset{\frown}{AD}=175^{\circ}$ and the minor arc be $m\overset{\frown}{AC}=85^{\circ}$. The formula for $\angle ABC$ is $\angle ABC=\frac{1}{2}(m\overset{\frown}{AD}-m\overset{\frown}{AC})$.

Step2: Substitute the arc - measures into the formula

Substitute $m\overset{\frown}{AD} = 175^{\circ}$ and $m\overset{\frown}{AC}=85^{\circ}$ into the formula: $\angle ABC=\frac{1}{2}(175 - 85)$.

Step3: Calculate the value

First, calculate the difference inside the parentheses: $175-85 = 90$. Then, multiply by $\frac{1}{2}$: $\frac{1}{2}\times90=45$.

Answer:

$45$