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

Correct choices:
- ∠EFD ≅ ∠EGD
- Arc E D ≅ Arc F D
- mArc F D = 120°

Reasoning (brief):
- The angle between the two tangents equals 180° − (minor arc ED). So 60° = 180° − m(arc ED) ⇒ m(arc ED) = 120°.
- Arcs EF and FD are given equal and together make the rest of the circle (the major arc ED = 240°), so each is 240°/2 = 120°. Hence mArc FD = 120° (and arc ED as the minor arc is 120°, so arc ED ≅ arc FD).
- The inscribed angle ∠EFD intercepts the minor arc ED, so ∠EFD = 1/2·120° = 60°, which equals ∠EGD.
- Therefore ∠EGD ≅ ∠ECD is false (∠ECD = central angle = 120°), and mArc EF = 60° is false (mArc EF = 120°).