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

1, 3, and 5.

i just got it right on the test.

Answer 2

• ∠EFD ≅ ∠EGD
• Arc E D is-congruent-to arc F D
• mArc F D = 120°

Explanation:

In Quadrilateral GECD,

`60^0+90^0+90^0+ \angle ECD=360^0\angle ECD=360^0-240^0=120^0\\\angle EFD=(1)/(2) \angle ECD \text{ (Inscribed Angle Theorem)}\\\angle EFD =(1)/(2) X 120 =60^0\\Therefore: \angle EFD=\angle EGD`

In Triangle EFD,

`\angle FED =\angle FDE \text{ (base angles of an isosceles triangle)}\\\angle FED +\angle FDE+\angle EFD=180^0\\\angle FED +\angle FDE+60=180\\\angle FED +\angle FDE=120\\\angle FED =\angle FDE=60^0`

Similarly, In Triangle GED

`\angle GED =\angle GDE \text{ (tangent to a circle)}\\\angle GED +\angle GDE+\angle EGD=180^0\\\angle GED +\angle GDE+60=180\\\angle GED +\angle GDE=120\\\angle GED =\angle GDE=60^0\\Therefore, mArc F D = 120\°`

Finally, Arc ED is-congruent-to arc F D

The first, third and last options are true.