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 ( angle abc )?
( mangle abc=)

Explanation:

Step1: Find the measure of the major arc

The sum of arcs in a circle is \(360^{\circ}\). Given arcs \(100^{\circ}\), \(175^{\circ}\), and \(85^{\circ}\).
The measure of the arc opposite to the angle formed by the tangent and secant:
Let the arc be \(x\).
\(x = 360-(100 + 175+85)\) is incorrect.
The formula for the angle formed by a tangent and a secant is \(\angle ABC=\frac{1}{2}(\text{major arc}-\text{minor arc})\).
The major arc \(= 175 + 85=260^{\circ}\), the minor arc \(=100^{\circ}\)

Step2: Apply the tangent - secant angle formula

The formula for the angle \(\theta\) formed by a tangent and a secant is \(\theta=\frac{1}{2}(m\overset{\frown}{AC}-m\overset{\frown}{AD})\) (where \(m\overset{\frown}{AC}\) is the measure of the major arc and \(m\overset{\frown}{AD}\) is the measure of the minor arc intercepted by the angle).
\(\angle ABC=\frac{1}{2}(260 - 100)\)

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Answer:

\(30\)