Question

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

Explanation:

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

The measure of an angle formed by two secants intersecting outside a circle is given by \(m\angle EPF=\frac{1}{2}(m\overarc{CG}-m\overarc{FG})\).

Step2: Substitute the known values into the formula

We know that \(m\angle EPF = 119^{\circ}\) and \(m\overarc{CG}=148^{\circ}\). Let \(m\overarc{FG}=x\). Then the equation becomes \(119=\frac{1}{2}(148 - x)\).

Step3: Solve the equation for \(x\)

Multiply both sides of the equation by \(2\): \(2\times119=148 - x\), so \(238 = 148 - x\).
Add \(x\) to both sides: \(238+x=148\).
Subtract \(148\) from both sides: \(x=148 - 238\), \(x = 106^{\circ}\).

Answer:

B. \(106^{\circ}\)