Question

points e, f, and d are on circle c, and angle g measures 60°. the measure of arc ef equals the measure of arc fd. which statements about the arcs and angles are true? select three options. ∠efd ≅ ∠egd ∠egd ≅ ∠ecd ed ≅ fd mef = 60° mfd = 120°

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Step2: Analyze \(\angle EFD\) and \(\angle EGD\)

\(\angle EFD\) and \(\angle EGD\) intercept the same arc \(\overset{\frown}{ED}\). By the inscribed - angle theorem, \(\angle EFD=\frac{1}{2}\overset{\frown}{ED}\) and \(\angle EGD=\frac{1}{2}\overset{\frown}{ED}\), so \(\angle EFD\cong\angle EGD\).

Step3: Analyze \(\angle EGD\) and \(\angle ECD\)

\(\angle EGD\) is an inscribed angle and \(\angle ECD\) is a central angle intercepting the same arc \(\overset{\frown}{ED}\). The measure of \(\angle ECD = 2\angle EGD\), so \(\angle EGD
ot\cong\angle ECD\).

Step4: Analyze \(\overline{ED}\) and \(\overline{FD}\)

Just because the measures of arcs \(\overset{\frown}{EF}\) and \(\overset{\frown}{FD}\) are equal does not mean the chords \(\overline{ED}\) and \(\overline{FD}\) are equal. Chord lengths depend on the radius and the central - angle subtended by the chord.

Step5: Analyze \(m\overset{\frown}{EF}\)

Let \(m\overset{\frown}{EF}=x\) and \(m\overset{\frown}{FD}=x\) (since \(m\overset{\frown}{EF}=m\overset{\frown}{FD}\)). The measure of the inscribed angle \(\angle EGD = 60^{\circ}\), and the measure of the central angle \(\angle ECD\) that intercepts the same arc as \(\angle EGD\) is \(120^{\circ}\). The sum of the measures of arcs \(\overset{\frown}{EF}\), \(\overset{\frown}{FD}\) and the remaining arc is \(360^{\circ}\). Since \(\angle EGD = 60^{\circ}\) and its central - angle counterpart for the arc it intercepts is \(120^{\circ}\), and \(m\overset{\frown}{EF}=m\overset{\frown}{FD}\), we know that \(m\overset{\frown}{EF}=m\overset{\frown}{FD}=60^{\circ}\).

Step6: Analyze \(m\overset{\frown}{FD}\)

As shown above, \(m\overset{\frown}{FD}=60^{\circ}
eq120^{\circ}\).

Answer:

A. \(\angle EFD\cong\angle EGD\)
D. \(m\overset{\frown}{EF}=60^{\circ}\)
E. \(m\overset{\frown}{FD}=60^{\circ}\) (Note: In the original options, the correct statements are \(\angle EFD\cong\angle EGD\), \(m\overset{\frown}{EF}=60^{\circ}\), and since \(m\overset{\frown}{EF} = m\overset{\frown}{FD}\), \(m\overset{\frown}{FD}=60^{\circ}\))