Angle G is a circumscribed angle of circle E. Major arc FD measures 280°.Circle E is shown. Line segments FE and DE are radii. A line is drawn to connect points F and D. Tangents FG and DG intersect at point G outside of the circle. Major arc FD measures 280 degrees. What is the measure of angle GFD?

finding the minor arc given the major arc

Solution:

From the figure, 

Major angle FED + Minor angle FED = 360°

280° + Minor angle FED = 360°

Minor angle FED = 360° - 280° = 80°

Also, ∠FED + ∠FGD = 180°

80° + ∠FGD = 180°

∠FGD = 180° - 80°

∠FGD = 100°

Now, GF and GD are the tangents to the circle from the same point G. Hence,

GD = GF

∴ ∠FDG = ∠GFD = x

(As these are the angles opposite to equal sides)

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

∴ ∠FDG + ∠FGD + ∠GFD = 180°

x + 100° + x = 180°

2x = 180° - 100°

2x = 80°

x = 40°

Angle G is a circumscribed angle of circle E. Major arc FD measures 280°.Circle E is shown. Line segments FE and DE are radii. A line is drawn to connect points F and D. Tangents FG and DG intersect at point G outside of the circle. Major arc FD measures 280 degrees. What is the measure of angle GFD?

Summary:

Given that angle G is a circumscribed angle of circle E. Major arc FD measures 280°.Circle E is shown. Line segments FE and DE are radii. A line is drawn to connect points F and D. Tangents FG and DG intersect at point G outside of the circle. Major arc FD measures 280 degrees. The measure of angle GFD`is 40°.

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