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To prove that \(\angle 1 = \angle 3\), we can construct a two-column proof with "Statements" and "Reasons." Let's analyze the given information and write the proof.

### Given:
- \( \overrightarrow{EB} \) bisects \(\angle AEC\)
- \( \overrightarrow{EC} \) bisects \(\angle BED\)

### To prove:
- \( \angle 1 = \angle 3 \)

### Proof Outline:

#### Statements | Reasons
1. \( \overrightarrow{EB} \) bisects \(\angle AEC\) | Given  
2. \( \overrightarrow{EC} \) bisects \(\angle BED\) | Given  
3. \( \angle 1 = \angle 2 \) | Definition of angle bisector (from statement 1)  
4. \( \angle 2 = \angle 3 \) | Definition of angle bisector (from statement 2)  
5. \( \angle 1 = \angle 3 \) | Transitive property of equality (from statements 3 and 4)

### Explanation:
- **Step 1 and 2:** These come directly from the problem's given conditions, that the rays \( \overrightarrow{EB} \) and \( \overrightarrow{EC} \) bisect their respective angles.
- **Step 3 and 4:** The definition of an angle bisector tells us that if a ray bisects an angle, it divides that angle into two equal parts. Hence, \( \angle 1 = \angle 2 \) and \( \angle 2 = \angle 3 \).
- **Step 5:** Since \( \angle 1 = \angle 2 \) and \( \angle 2 = \angle 3 \), by the transitive property, we can conclude \( \angle 1 = \angle 3 \).

Would you like to go into more details on any specific part of this proof?

Here are 5 questions that might help you expand on this concept:

1. What is the transitive property, and how is it applied in geometric proofs?
2. How would the proof change if \( \overrightarrow{EB} \) and \( \overrightarrow{EC} \) were not bisectors?
3. Can angle bisectors be used in solving problems involving triangles? If so, how?
4. How do parallel lines influence angle relationships in geometric proofs?
5. What role do congruent angles play in proving that two lines are parallel?

**Tip:** Always remember that the transitive property is a powerful tool in proofs when dealing with equal angles or segments.

Prove that angle 1 equals angle 3 using the bisectors of angles AEC and BED.

Learn how to prove that angle 1 equals angle 3 using angle bisectors and the transitive property. This geometric proof involves bisectors of angles AEC and BED and applies the definition of an angle bisector to conclude that two angles are equal.

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