Question

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

Explanation:

Step1: Recall the formula for the measure of an angle formed by two intersecting chords

The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs. So, \(m\angle EPF=\frac{1}{2}(m\overset{\frown}{CD}+m\overset{\frown}{FG})\)

Step2: Substitute the known values into the formula

We know that \(m\angle EPF = 116^{\circ}\) and \(m\overset{\frown}{CD}=86^{\circ}\). Substituting these values into the formula \(116^{\circ}=\frac{1}{2}(86^{\circ}+m\overset{\frown}{FG})\)

Step3: Solve for \(m\overset{\frown}{FG}\)

First, multiply both sides of the equation by \(2\): \(2\times116^{\circ}=86^{\circ}+m\overset{\frown}{FG}\)
\(232^{\circ}=86^{\circ}+m\overset{\frown}{FG}\)
Then, subtract \(86^{\circ}\) from both sides: \(m\overset{\frown}{FG}=232^{\circ}- 86^{\circ}\)
\(m\overset{\frown}{FG}=146^{\circ}\)

Answer:

D. \(146^{\circ}\)