**Tangents $\text DE$ and $\text DF$ are drawn to a circle from an external point $\text D$. The angle $\angle DFE = 50 \degree$. A chord $\text GH$ is drawn parallel to the tangent $\text DE$. Find the measure of $\angle GDF$.** 

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} $.

---

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.

Tangents DE and DF are drawn to a circle from an external point D. The angle ∠DFE = 50°. A chord GH is drawn parallel to the tangent DE. Find the measure of ∠GDF.

Explore how to find the measure of ∠GDF using tangents, a chord parallel to the tangent, and the Alternate Interior Angle Theorem. With the given angle ∠DFE = 50°, this geometry problem covers core circle theorems and tangent properties, suitable for students in Grades 10-12.

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