Question

select the correct answer. in circle c, m\\(widehat{cg}=148^{circ}\\), m\\(widehat{cd}=86^{circ}\\), and m\\(\angle eff = 116^{circ}\\). what is the measure of arc fg? figure not drawn to scale. a. 86° b. 106° c. 116° d. 146°

Explanation:

Step1: Recall the angle - arc relationship

The measure of an exterior angle of a circle formed by two secants is given by $\frac{1}{2}(m\overparen{long\ arc}-m\overparen{short\ arc})$. Here, $\angle EPF$ is an exterior angle formed by secants $CEP$ and $GEP$. Let $m\overparen{FG}=x$. The long - arc is $\overparen{CG}$ and the short - arc is $\overparen{CD}$.

Step2: Set up the equation

We know that $m\angle EPF = 116^{\circ}$, $m\overparen{CG}=148^{\circ}$, and $m\overparen{CD}=86^{\circ}$. Using the formula $m\angle EPF=\frac{1}{2}(m\overparen{CG}-m\overparen{CD}+m\overparen{FG})$. Substitute the known values into the formula: $116=\frac{1}{2}(148 - 86+x)$.

Step3: Solve the equation

First, simplify the expression inside the parentheses: $148−86 = 62$. So the equation becomes $116=\frac{1}{2}(62 + x)$. Multiply both sides of the equation by 2 to get rid of the fraction: $2\times116=62 + x$. Then $232=62 + x$. Subtract 62 from both sides: $x=232 - 62=106^{\circ}$.

Answer:

B. $106^{\circ}$