Question

points e, f, and d are on circle c, and angle g measures $60^{\circ}$. the measure of arc ef equals the measure of arc fd.
which statements about the arcs and angles are true? select three options.
$\angle efd \cong \angle egd$
$\angle egd \cong \angle ecd$
$\overparen{ed} \cong \overparen{fd}$
$m\overparen{ef} = 60^{\circ}$
$m\overparen{fd} = 120^{\circ}$

Explanation:

Step1: Analyze cyclic quadrilateral

Quadrilateral \(ECDF\) is cyclic, so \(\angle ECD + \angle EFD = 180^\circ\). Also, \(GE \perp EC\) and \(GD \perp DC\), so \(\angle GEC = \angle GDC = 90^\circ\). In quadrilateral \(GECD\), \(\angle EGD + \angle ECD = 360^\circ - 90^\circ - 90^\circ = 180^\circ\). Thus \(\angle EFD \cong \angle EGD\).

Step2: Verify arc congruence

Given \(m\overset{\frown}{EF} = m\overset{\frown}{FD}\). Chords \(ED\) and \(FD\) are equal (from congruent arcs), so \(\overset{\frown}{ED} \cong \overset{\frown}{FD}\) is false.

Step3: Calculate arc measures

In quadrilateral \(GECD\), \(\angle EGD = 60^\circ\), so \(\angle ECD = 120^\circ\) (supplementary). \(\angle ECD\) is a central angle for \(\overset{\frown}{ED}\), so \(m\overset{\frown}{ED} = 120^\circ\). The total circle is \(360^\circ\), so \(m\overset{\frown}{EF} + m\overset{\frown}{FD} = 360^\circ - 120^\circ = 240^\circ\). Since \(m\overset{\frown}{EF} = m\overset{\frown}{FD}\), each is \(120^\circ\), so \(m\overset{\frown}{FD}=120^\circ\) is true, \(m\overset{\frown}{EF}=60^\circ\) is false.

Step4: Check angle congruence

\(\angle EGD = 60^\circ\), \(\angle ECD = 120^\circ\), so \(\angle EGD
ot\cong \angle ECD\).

Answer:

• $\angle EFD \cong \angle EGD$
• $m\overset{\frown}{FD} = 120^\circ$