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? choose three correct answers.
$ \overleftrightarrow { e d } \cong \overleftrightarrow { f d } $
$ \angle e f d \cong \angle e g d $
$ m \overleftrightarrow { e f } = 60 ^ { \circ } $
$ \angle e g d \cong \angle e c d $
$ m \overleftrightarrow { f d } = 120 ^ { \circ } $

Question

Accepted Answer

# Explanation:
## Step1: Analyze the quadrilateral $EGDC$
Since $ \angle GED=\angle GDC = 90^{\circ}$ (tangent - radius property) and $ \angle EGD=60^{\circ}$, using the sum of angles in a quadrilateral ($360^{\circ}$), we can find $ \angle ECD$.
$$
\begin{align*}
\angle EGD+\angle GED+\angle GDC+\angle ECD&=360^{\circ}\
60^{\circ}+90^{\circ}+90^{\circ}+\angle ECD&=360^{\circ}\
\angle ECD&=120^{\circ}
\end{align*}
$$
Since $CE = CF = CD$ (radii of the circle) and $ \overset{\frown}{EF}=\overset{\frown}{FD}$, let $m\overset{\frown}{EF}=m\overset{\frown}{FD}=x$. Then $m\overset{\frown}{ED}=2x$ (because $ \angle ECD = 120^{\circ}$ and the central angle is equal to the measure of its arc).

## Step2: Use the property of inscribed angles and arcs
The measure of an inscribed angle is half the measure of its intercepted arc.
- For the first option: Since $CE = CD$ (radii) and $ \angle ECD = 120^{\circ}$, $ \overset{\frown}{ED}$ is not congruent to $ \overset{\frown}{FD}$ (because $m\overset{\frown}{ED} = 120^{\circ}$ and $m\overset{\frown}{FD}=60^{\circ}$).
- For the second option: The measure of $ \angle EGD = 60^{\circ}$. The measure of $ \angle EFD$: Since $ \overset{\frown}{ED}=120^{\circ}$, the inscribed angle $ \angle EFD=\frac{1}{2}m\overset{\frown}{ED}$ (inscribed - arc relationship). So $m\angle EFD = 60^{\circ}$, and $ \angle EFD\cong\angle EGD$.
- For the third option: Since $ \angle ECD=120^{\circ}$ and $m\overset{\frown}{EF}+m\overset{\frown}{FD}=m\overset{\frown}{ED}$ (arc addition) and $m\overset{\frown}{EF}=m\overset{\frown}{FD}$, $m\overset{\frown}{EF} = 60^{\circ}$.
- For the fourth option: $ \angle EGD = 60^{\circ}$ and $ \angle ECD=120^{\circ}$, so $ \angle EGD
ot\cong\angle ECD$.
- For the fifth option: $m\overset{\frown}{FD}=60^{\circ}$ (because $m\overset{\frown}{EF}=m\overset{\frown}{FD}$ and $m\overset{\frown}{EF}+m\overset{\frown}{FD}=m\overset{\frown}{ED}$ with $m\overset{\frown}{ED} = 120^{\circ}$)

# Answer:
$\angle EFD\cong\angle EGD$, $m\overset{\frown}{EF}=60^{\circ}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Analyze the quadrilateral \(EGDC\)

Step2: Use the property of inscribed angles and arcs

Answer: