Question

Angle G is a circumscribed angle of circle E. Major arc FD measures 280°.

Circle E is shown. Line segments F E and D E are radii. A line is drawn to connect points F and D. Tangents F G and D G intersect at point G outside of the circle. Major arc F D measures 280 degrees.

What is the measure of angle GFD?
40°
50°
80°
90

Answer 1

The measure of angle GFD of the circumscribed circle is; A: 40°

How to find the measure of angle of a circumscribed circle?

From the figure, we can apply the arc angles summation formula to get;

Major angle ∠FED + Minor angle ∠FED = 360°

We are given that Major arc FD measures 280°. Thus;

280° + Minor angle ∠FED = 360°

Minor angle ∠FED = 360° - 280°

Minor angle ∠FED = 80°

Also, we know that;

∠FED + ∠FGD = 180°

Thus, putting ∠FED = 80° gives us;

80° + ∠FGD = 180°

Subtract 80° from both sides using subtraction property of equality to get;

∠FGD = 180° - 80°

∠FGD = 100°

Now, GF and GD are the tangents to the circle from the same point G. Thus, we can say that;

GD = GF

Therefore,

∠FDG = ∠GFD = x

(This is because ∠FDG and ∠GFD are the angles opposite to equal sides.

In triangle FGD, we have sum of interior angles = 180°

Therefore, we have the expression;

∠FDG + ∠FGD + ∠GFD = 180° (due to the fact that sum of angles in a triangle is equal to 180°)

x + 100° + x = 180°

2x = 180° - 100°

2x = 80°

x = 40°

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