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

Work out the arcs from the tangent-angle fact. For two tangents the outside angle = 180° − (measure of the minor intercepted arc). So

60° = ∠EGD = 180° − m(arc ED) ⇒ m(arc ED) = 120°.

Given arc EF = arc FD, let each equal x. Then arc ED = arc EF + arc FD = 2x = 120°, so x = 60°. Thus arc EF = 60°, arc FD = 60°, arc ED = 120°.

Now check the statements:

- ∠EFD ≅ ∠EGD — True. ∠EFD is an inscribed angle intercepting arc ED, so it measures 1/2·120° = 60°, equal to ∠EGD.
- ∠EGD ≅ ∠ECD — False. ∠ECD is a central angle equal to arc ED = 120°, not 60°.
- Arc ED ≅ arc FD — False. Arc ED = 120°, arc FD = 60°.
- mArc EF = 60° — True.
- mArc FD = 120° — False (mArc FD = 60°).

So the true statements are: ∠EFD ≅ ∠EGD and mArc EF = 60°. (Only two of the given statements are correct.)