Question

question 2 of 10
in the diagram below, \\( \overline { d e } \\) and \\( \overline { e f } \\) are tangent to \\( \odot o \\). what is the measure of
\\( \overparen { d g f } \\) ?
a. \\( 195 ^ { \circ } \\)
b. \\( 245 ^ { \circ } \\)
c. \\( 100 ^ { \circ } \\)
d. \\( 180 ^ { \circ } \\)

Explanation:

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

The measure of an angle formed by two tangents \( \angle E\) and the sum of the measures of the intercepted arcs \( \overset{\frown}{DGF}\) and \( \overset{\frown}{DF}\) is given by the formula \( \angle E=\frac{1}{2}(m\overset{\frown}{DGF}-m\overset{\frown}{DF})\). Also, we know that \(m\overset{\frown}{DGF}+m\overset{\frown}{DF} = 360^{\circ}\), so \(m\overset{\frown}{DF}=360^{\circ}-m\overset{\frown}{DGF}\).

Step2: Substitute into the angle - tangent formula

Given \( \angle E = 65^{\circ}\), substitute \(m\overset{\frown}{DF}=360 - m\overset{\frown}{DGF}\) into \( \angle E=\frac{1}{2}(m\overset{\frown}{DGF}-m\overset{\frown}{DF})\). Then \(65^{\circ}=\frac{1}{2}(m\overset{\frown}{DGF}-(360^{\circ}-m\overset{\frown}{DGF}))\).

Step3: Simplify the equation

First, expand the right - hand side: \(65^{\circ}=\frac{1}{2}(m\overset{\frown}{DGF}- 360^{\circ}+m\overset{\frown}{DGF})\). Combine like terms: \(65^{\circ}=\frac{1}{2}(2m\overset{\frown}{DGF}-360^{\circ})\). Multiply both sides by 2: \(130^{\circ}=2m\overset{\frown}{DGF}-360^{\circ}\).

Step4: Solve for \(m\overset{\frown}{DGF}\)

Add \(360^{\circ}\) to both sides: \(130^{\circ}+ 360^{\circ}=2m\overset{\frown}{DGF}\), so \(490^{\circ}=2m\overset{\frown}{DGF}\). Divide both sides by 2: \(m\overset{\frown}{DGF}=245^{\circ}\).

Answer:

B. \(245^{\circ}\)