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This problem is a geometric proof involving angle bisectors and congruence of angles. Let's break it down and build the two-column proof (Statements and Reasons).

### **Given:**
1. \( OB \) bisects \( \angle AOC \).
2. \( OE \) bisects \( \angle DOF \).
3. \( \angle AOB \cong \angle DOE \).

### **To Prove:**
\[
\angle EOF \cong \angle BOC
\]

### **Diagram Analysis:**
- Line segments \( OB \), \( OE \), and others intersect at point \( O \), creating several angles.
- \( OB \) bisects \( \angle AOC \), which means \( \angle AOB = \angle BOC \).
- \( OE \) bisects \( \angle DOF \), meaning \( \angle DOE = \angle EOF \).
- Given \( \angle AOB \cong \angle DOE \), we need to prove \( \angle EOF \cong \angle BOC \).

### **Proof:**

| **Statements**                        | **Reasons**                                            |
|---------------------------------------|--------------------------------------------------------|
| 1. \( OB \) bisects \( \angle AOC \)  | Given                                                  |
| 2. \( OE \) bisects \( \angle DOF \)  | Given                                                  |
| 3. \( \angle AOB \cong \angle DOE \)  | Given                                                  |
| 4. \( \angle AOB = \angle BOC \)      | Definition of angle bisector (from statement 1)         |
| 5. \( \angle DOE = \angle EOF \)      | Definition of angle bisector (from statement 2)         |
| 6. \( \angle AOB \cong \angle DOE \)  | Given                                                  |
| 7. \( \angle BOC \cong \angle EOF \)  | Substitution property (from steps 4 and 5)              |
| 8. \( \angle EOF \cong \angle BOC \)  | By transitivity (from steps 6 and 7)                    |

### Conclusion:
Thus, we have proven that \( \angle EOF \cong \angle BOC \), as required.

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Would you like further details on any part of this solution?

Here are five related questions to explore further:

1. How do you define an angle bisector?
2. What properties of angle bisectors are used in proving congruence?
3. Can this proof be extended if more angles are bisected in the diagram?
4. Why is the substitution property key in geometric proofs?
5. What role does transitivity play in proving congruent angles?

**Tip:** In geometry, always ensure you clearly define every element in the diagram, as missing details can lead to incorrect conclusions!

Given that OB bisects ∠AOC and OE bisects ∠DOF, and ∠AOB ≅ ∠DOE, prove that ∠EOF ≅ ∠BOC.

This problem involves using angle bisectors and the congruence of angles to prove that ∠EOF ≅ ∠BOC. Given that OB and OE bisect the angles, and ∠AOB ≅ ∠DOE, this proof applies geometric principles such as the Angle Bisector Theorem and the Transitive Property. Suitable for students in Grades 9-10.

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