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The problem presented asks about the angles of the paths and includes certain conditions:

1. **Path \( EF \) is longer than path \( ED \)**.
2. **Path \( GD \) is equal in length to path \( GF \)**.

Given these details, you are asked to conclude how the angles compare, focusing on angle relationships.

### Analysis:

1. **Triangle Inequality Theorem**: In any triangle, the longer the side, the larger the opposite angle.
   - Since \( EF > ED \), angle \( \angle EFG \) (opposite the longer side \( EF \)) must be larger than angle \( \angle EDG \) (opposite the shorter side \( ED \)).
   
2. **Congruent paths**: Since \( GD = GF \), triangle \( GDF \) is isosceles. Therefore, angles \( \angle EGF \) and \( \angle EGD \) are equal.

### Conclusion:

The correct answer is: 
- \( m\angle EFG > m\angle EDG \), because the opposite sides to these angles indicate this relationship based on the length of the paths.

Would you like to explore more details or ask further questions about this problem?

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Here are some related questions you might find useful:

1. How does the triangle inequality theorem help in comparing angles?
2. What happens to the angles in an isosceles triangle when two sides are equal?
3. How would this situation change if \( GD \) were longer than \( GF \)?
4. What additional geometric rules apply when comparing angles in different triangles?
5. How could you verify the angle comparison using a protractor or other geometric tools?

**Tip**: Always remember to relate the size of angles to the opposite side lengths in a triangle. The longer the side, the larger the opposite angle!

The path from E to F is longer than the path from E to D. The path from G to D is the same length as the path from G to F. What can you conclude about the angles of the paths?

This problem demonstrates how the Triangle Inequality Theorem is applied to compare angles based on the length of paths in a triangle. It involves paths from E to F and E to D, as well as a comparison of angles using isosceles triangle properties. Suitable for students in Grades 8-10.

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