We are given a circle with the following information:

• $DE$ and $DF$ are tangents from an external point $D$ to the circle.
• $\angle DFE = 50^\circ$.
• A chord $GH$ is drawn parallel to the tangent $DE$. We are tasked with finding the measure of $\angle GDF$.

Step 1: Properties of the tangents and the angle $\angle DFE$

From the properties of tangents, we know that the tangents drawn from an external point to a circle are equal in length. Therefore, $DE = DF$. Additionally, since $DE$ and $DF$ are tangents, $\angle DEF$ is an isosceles triangle.

Now, $\angle DFE = 50^\circ$. By symmetry of the tangents, $\angle DFE = \angle DEF$, so:

$\angle DEF = 50^\circ.$

Thus, the total angle at the external point $D$ is:

$\angle EDF = 180^\circ - \left( \angle DFE + \angle DEF \right) = 180^\circ - (50^\circ + 50^\circ) = 80^\circ.$

Step 2: Parallel lines and the angle $\angle GDF$

We are given that the chord $GH$ is parallel to the tangent $DE$. Using the fact that tangents and parallel lines create alternate interior angles, we can say that:

$\angle GDF = \angle DEF = 50^\circ.$

Conclusion:

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

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Would you like further details or have any questions? Here are some related questions to deepen your understanding:

1. What are the properties of tangents to a circle?
2. How do we prove that tangents drawn from an external point are equal?
3. What is the relationship between parallel lines and alternate interior angles?
4. Can you explain why $\angle EDF$ is 80° in this problem?
5. How would the answer change if $GH$ were not parallel to $DE$?

Tip: When working with tangents and circles, remember that angles formed by tangents and chords often involve symmetrical relationships and alternate angle properties.