https://file.aiquickdraw.com/mathbot/file/1726194199364dklk8h6u.jpg

To complete the proof that \(\angle EDF \cong \angle GDH\), let's analyze the given information and fill in the missing statements and reasons.

### Given Information:
1. **\( \overrightarrow{DF} \) bisects \(\angle EDG\) and \( \overrightarrow{DG} \) bisects \(\angle FDH\).** (Given)

### To Prove:
\(\angle EDF \cong \angle GDH\)

### Completing the Proof:

Let's fill in the statements and reasons step by step:

1. **Statement**: \( \overrightarrow{DF} \) bisects \(\angle EDG\) and \( \overrightarrow{DG} \) bisects \(\angle FDH\).  
   **Reason**: Given.

2. **Statement**: \(\angle EDF \cong \angle FDG\)  
   **Reason**: Definition of bisector.  
   - Since \( \overrightarrow{DF} \) bisects \(\angle EDG\), it divides it into two equal angles: \(\angle EDF \cong \angle FDG\).

3. **Statement**: \(\angle GDH \cong \angle FDG\)  
   **Reason**: Definition of bisector.  
   - Since \( \overrightarrow{DG} \) bisects \(\angle FDH\), it divides it into two equal angles: \(\angle GDH \cong \angle FDG\).

4. **Statement**: \(\angle EDF \cong \angle GDH\)  
   **Reason**: Transitive property of congruence.  
   - Since \(\angle EDF \cong \angle FDG\) and \(\angle FDG \cong \angle GDH\), by the transitive property, \(\angle EDF \cong \angle GDH\).

### Conclusion:
Thus, the proof is completed, showing that \(\angle EDF \cong \angle GDH\).

Would you like more details, or do you have any questions? 

### Related Questions to Expand Understanding:
1. What is the definition of an angle bisector in geometry?
2. How does the reflexive property of congruence apply to geometric proofs?
3. Can you provide an example of a different geometric proof involving angle bisectors?
4. How does the transitive property of congruence work in other mathematical contexts?
5. What are some common properties of angles and their uses in geometric proofs?

### Tip:
When solving geometric proofs, always carefully analyze the given information and properties, such as congruence, bisectors, and reflexive properties, to build your logical steps clearly.

Use the information to complete the proof. In the figure, DF bisects ∠EDG and DG bisects ∠FDH.

Learn how to complete a geometric proof involving bisectors of angles. This problem involves concepts like the definition of bisectors, the transitive and reflexive properties of congruence, suitable for students in Grades 9-10.

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