Question

question 5 of 10
if the measure of \\( \angle d e f \\) is \\( 65^{circ} \\), what is the measure of \\( \overparen{d e} \\) ?
a. \\( 130^{circ} \\)
b. \\( 32.5^{circ} \\)
c. \\( 115^{circ} \\)
d. \\( 65^{circ} \\)

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant is half the measure of the intercepted arc. The formula is \(m\angle DEF=\frac{1}{2}m\overarc{DE}\).

Step2: Solve for \(m\overarc{DE}\)

Given \(m\angle DEF = 65^{\circ}\), from the formula \(m\angle DEF=\frac{1}{2}m\overarc{DE}\), we can solve for \(m\overarc{DE}\) by multiplying both sides of the equation by \(2\). So \(m\overarc{DE}=2\times m\angle DEF\).
Substitute \(m\angle DEF = 65^{\circ}\) into the equation: \(m\overarc{DE}=2\times65^{\circ}\).

Answer:

A. \(130^{\circ}\)