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.

Circle C is shown. Line segments E C and D F are radii. Lines are drawn from points E and D to point F to form chords E F and D F. Tangents E G and D G intersect at point G outside of the circle. Angle E G D is 60 degrees and angles G E C and G D C are right angles. The lengths of E F and D F are congruent.


Which statements about the arcs and angles are true? Select three options.


∠EFD ≅ ∠EGD

∠EGD ≅ ∠ECD

Arc E D is-congruent-to arc F D

mArc E F = 60°

mArc F D = 120°

Answer 1

∠EFD ≅ ∠EGD ⇒ A

Arc ED ≅ arc FD ⇒ C

m arc FD = 120° ⇒ E

Explanation:

Let us revise some facts

1. Equal chords subtended equal arcs
2. The measure of an inscribed angle is one-half the measure of the central angle which subtended by the same arc
3. The measure of a central angle is equal to the measure of its subtended arc
4. If one angle of an isosceles triangle measure 60° then the triangle is equilateral
5. The sum of the measures of the interior angles of any quadrilateral is 360°

In the quadrilateral CDGE

∵ m∠G = 60°

∵ m∠GDC = m∠GEC = 90°

- By using the 5th rule above

∴ m∠G + m∠GDC + m∠DCE + m∠GEC = 360°

∴ 60 + 90 + m∠DCE + 90 = 360

∴ 240 + m∠DCE = 360

- Subtract 240 from both sides

∵ m∠DCE = 120°

In circle C

∵ ∠DCE is a central angle subtended by arc DE

∵ ∠DFE is an inscribed angle subtended by arc DE

- By using the 2nd rule above

∴ m∠DFE =
`(1)/(2)` m∠∠DCE

∵ m∠DCE = 120°

∴ m∠DFE =
`(1)/(2)` (120)

∴ m∠DFE = 60°

- That means ∠EFD ≅ ∠EGD because their measure is 60°

∴ ∠EFD ≅ ∠EGD

In Δ EFD

∵ EF = FD

∵ m∠DFE = 60°

- By using the 4th rule above

∴ Δ EFD is an equilateral triangle

∴ ED = FD = FE

In circle C

∵ Side ED subtended by arc ED

∵ Side FD subtended by FD

∵ Side ED ≅ side FD ⇒ proved

- By using the 1st rule above

∴ Arc ED ≅ arc FD

∵ m∠ECD = 120°

∵ ∠ECD is a central angle subtended by arc ED

- By using the 3rd rule above

∴ m∠ECD = m arc ED

∴ m of arc ED = 120°

∵ Arc ED ≅ arc FD

∴ m arc ED = m arc FD

∴ m arc FD = 120°

Answer 2

Angle EFD is congruent to angle EGD, both measuring 60 degrees, as radii EF and FD bisect the circle. Angle EGD is also congruent to ∠ECD, forming a true statement. Additionally, m⌢FD equals 120° due to the given information. (A, B and E)

Let's analyze the given information:

1. Angle EGD is given as 60 degrees.

2. Angles GEC and GDC are right angles.

3. EF and DF are congruent.

Now, let's determine the true statements:

a. Angle EFD ≈ Angle EGD: This is indeed true. Angle EGD is given as 60 degrees, and angle EFD is formed by radii EF and FD, dividing the circle into two equal parts. Therefore, angle EFD is also 60 degrees.

b. Angle EGD ≈ ∠ECD: This is true. Angle EGD is given as 60 degrees, and ∠ECD is a right angle, so they are congruent.

c. ⌢ED = ⌢FD: This is not necessarily true. While EF and DF are congruent, it does not mean that the arcs themselves (⌢ED and ⌢FD) are congruent.

d. m⌢EF = 60°: This is not necessarily true. The measure of arc EF is not provided in the given information.

e. m⌢FD = 120°: This is true. Since angle EGD is given as 60 degrees, and angles GEC and GDC are right angles, the sum of angles GEF and GDF is 180 degrees. Therefore, the measure of arc FD is 120 degrees (180° - 60°).

So, the correct statements are:

a. Angle EFD ≈ Angle EGD

b. Angle EGD ≈ ∠ECD

e. m⌢FD = 120°