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°

The relationship between the angles formed in a circle are described by

circle theorems.

The true statements are;

m∠EFD ≅ m∠EGD ≅ = 120°

Reasons:

According to the outside angle theorem, we have;

Therefore;

2 × m∠EGD =

2 × 60° = 120° =

= 360° sum of angles formed at the center of the circle

= 360° -

120° =

120° = - (360° -

120° = 2· - 360°

Which gives;

60° = - 180°

= 60° + 180° = 240°

= 240°

= 360° - = 360° - 240° = 120°

= 120°

 = 2 × m∠EFD angle subtended by an arc at the center is twice the angle subtended at the circumference

∴  = 120° = 2 × m∠EFD

Therefore;

m∠EFD = m∠EGD = m∠G = 60°

m∠EGD = 60°

m∠EFD ≅ m∠EGD by definition of congruency

= m∠ECD by definition of the measure of an arc

∴ = 120° = m∠ECD

∴ m∠ECD ≠ m∠EGD = 60°

m∠ECD m∠EGD

Given that m∠EFD = 60°, and ΔEFD is an isosceles triangle, we have;

∠DEF = ∠FDE

∠DEF + ∠FDE = 180° - 60° = 120°

∴ ∠DEF = ∠FDE = 60°

Therefore;

ΔEFD is an equilateral triangle (all angles are equal to 60°)

= Equal chord subtend equal minor arcs theorem

Therefore;

by definition of congruency

= 120° =

∴

Learn more about circle theorems here:

link

∠EFD ≅ ∠EGD ⇒ A

Arc ED ≅ arc FD ⇒ C

m arc FD = 120° ⇒ E

Step-by-step explanation:

Let us revise some facts

Equal chords subtended equal arcsThe measure of an inscribed angle is one-half the measure of the central angle which subtended by the same arcThe measure of a central angle is equal to the measure of its subtended arcIf one angle of an isosceles triangle measure 60° then the triangle is equilateralThe 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 = m∠∠DCE

∵ m∠DCE = 120°

∴ m∠DFE = (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°

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