High School

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Points D, E, and F are on circle C, and EF ≅ DF.

Circle C is shown with line segments DC and EC as radii. Point F is on the opposite side of the circle.

Lines are drawn from points D and E to point F.

Tangents DG and EG intersect at point G.

Angle DGE is 76 degrees, and angles GDC and GEC are 90 degrees.

DF and EF are congruent.

Answer :

Final answer:

The question pertains to geometry involving circles, radii, and tangents. It provides an example scenario in which the properties of an isosceles triangle and tangents are interactively used to find angles in a given configuration.

Explanation:

This question seems to be on the topic of geometry, more specifically dealing with circles, radii, and tangents. Given that points D, E, and F are on circle C, and that EF is congruent to DF, we can infer certain properties. As lines DC and EC are radii, they will be of equal length. Because the given points form an isosceles triangle as DF and EF are congruent, the angles opposite them will be equal.

Now, coming to the tangent lines DG and EG, since they are tangents, they make a 90 degree angle with the radii at the points of tangency. Thus, given that the angle DGE is 76 degrees, using the properties of a triangle, the other two angles will be 180 -76 = 104 degrees.

This shows that angle DGF and EGF will be exactly half of 104 degrees, i.e., 52 degrees each. This is due to the fact that in an isosceles triangle, the base angles are equal. Therefore, the existence of an isosceles triangle DF and EF being congruent helps us determine the angles of the triangle.

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