Question

in the diagram shown below, \\( \overline { b f } \\) and \\( \overline { c b } \\) are tangent to the circle and \\( m \angle c b f = 8 2 ^ { \circ } \\).
what is the measure of \\( \overparen { c g f } \\)?
5 of 6 question
\\( 2 7 8 ^ { \circ } \\)
\\( 2 6 2 ^ { \circ } \\)
\\( 2 4 4 ^ { \circ } \\)
\\( 1 6 4 ^ { \circ } \\)

Explanation:

Step1: Recall the property of the angle formed by two tangents

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. Let the measure of the major arc \( \overarc{CGF}\) be \(x\) and the measure of the minor arc \( \overarc{CF}\) be \(y\). We know that \(x + y=360^{\circ}\) (since the sum of the measures of a major arc and its corresponding minor arc in a circle is \(360^{\circ}\)), and the measure of the angle \( \angle CBF\) formed by the two tangents is given by \(m\angle CBF=\frac{1}{2}(x - y)\).

Step2: Substitute the given value of the angle

We are given that \(m\angle CBF = 82^{\circ}\). So, \(82^{\circ}=\frac{1}{2}(x - y)\). Multiply both sides of the equation by \(2\) to get \(x - y=164^{\circ}\).

Step3: Solve the system of equations

We have the system of equations:
\(

$$\begin{cases}x + y=360^{\circ}\\x - y=164^{\circ}\end{cases}$$

\)
Add the two equations together: \((x + y)+(x - y)=360^{\circ}+164^{\circ}\).
This simplifies to \(2x=524^{\circ}\).
Divide both sides by \(2\) to find \(x\): \(x = 262^{\circ}\)

Answer:

\(262^{\circ}\)