**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 the following information:
- Tangents $ DE $ and $ DF $ are drawn to a circle from an external point $ D $.
- The angle $ \angle DFE = 50^\circ $.
- A chord $ GH $ is drawn parallel to the tangent $ DE $.
We are tasked with finding the measure of the angle $ \angle GDF $.

### Step 1: Analyzing the geometry of the problem
Since $ DE $ and $ DF $ are tangents from the same external point $ D $, the angles formed between the tangents and the radii at the points of contact are $ 90^\circ $. Therefore, the angle $ \angle EDF $ is an external angle formed by these tangents.

#### Important properties of tangents:
- The tangents drawn from a point outside the circle to the circle are equal in length. Hence, $ DE = DF $.
- The angles formed between the tangents and the chord from the point of contact follow certain relationships.

### Step 2: Using the given parallel condition
Since the chord $ GH $ is parallel to the tangent $ DE $, we can use the property of parallel lines intersected by a transversal. In this case, the transversal is the line segment $ DF $, which intersects both $ GH $ and $ DE $. By the alternate interior angle theorem, the angle $ \angle GDF $ is equal to the angle $ \angle DFE $.

### Step 3: Conclusion
Since $ \angle DFE = 50^\circ $, we have:

\[
\angle GDF = 50^\circ
\]

Thus, the measure of $ \angle GDF $ is $ \boxed{50^\circ} $.

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Would you like further details or clarifications on any steps?

Here are 5 related questions to deepen your understanding:
1. How do you prove that tangents drawn from an external point are equal in length?
2. Why are the angles between a tangent and a radius $ 90^\circ $?
3. What is the relationship between parallel lines and transversal angles in this context?
4. How would the answer change if $ GH $ were not parallel to $ DE $?
5. Can you generalize this problem to any angle between tangents and parallel chords?

**Tip:** Remember that tangents from a common external point have unique geometric properties that help in solving angle problems in circles.

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 angle ∠GDF in a circle where tangents DE and DF are drawn, and a chord GH is parallel to tangent DE. Using circle theorems and the alternate interior angle theorem, we solve this geometry problem step-by-step. Suitable for Grades 9-11.

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