We are given a circle with tangents $DE$ and $F$ drawn from a point $D$ such that $\angle DFE = 50^\circ$, and a chord $GH$ is drawn parallel to $DE$. We need to find the measure of $\angle GDF$.

Key Observations:

1. Since $DE$ and $F$ are tangents, we know the angle between the tangents at point $D$ will be equal to the angle between their respective radii at the point of contact, i.e., $\angle DFE$.

2. The chord $GH$ is parallel to the tangent $DE$. This implies that angles formed with the chord $GH$ inside the circle will be related to angles formed with the tangent $DE$ outside the circle by alternate interior angle relationships.

Steps to Solution:

1. Tangent-Chord Angle Theorem: According to this theorem, the angle between a tangent and a chord drawn from the point of contact is equal to the angle subtended by the chord on the opposite side of the circle. This means that the angle subtended by the chord $GH$ at the point $D$, i.e., $\angle GDF$, is equal to $\angle DFE$ because $GH \parallel DE$.

2. Given Information: $\angle DFE = 50^\circ$ Therefore, since $\angle GDF = \angle DFE$, we have: $\angle GDF = 50^\circ$

Final Answer:

The measure of $\angle GDF$ is $50^\circ$.

Would you like further details on any part of this explanation?

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Here are five related questions to extend this understanding:

1. What is the tangent-secant theorem and how does it apply to circles?
2. How can we prove that a tangent is perpendicular to the radius at the point of contact?
3. What is the relationship between angles subtended by the same chord in a circle?
4. How do parallel lines affect angles in a circle, particularly involving chords and tangents?
5. Can you find the length of the chord $GH$ if the radius of the circle and distance from $D$ to the center are known?

Tip: Always look for symmetries or parallelism in circle geometry problems as they often simplify angle relationships.