https://file.aiquickdraw.com/mathbot/file/1727657851026z50fum0t.jpg

Let's work through this problem step by step by completing the proof and specifying the correct reasons for each step. Here's the image's content analyzed in detail:

### **Given Information:**
- \( m \angle AXC = m \angle DYF \)
- \( m \angle 2 = m \angle 4 \)
  
### **Prove:**
- \( m \angle 1 = m \angle 3 \)

### **Proof Table Analysis:**

The proof table contains **Statements** and corresponding **Reasons**. We need to determine the correct reason for each step.

### **Statements and Corresponding Reasons:**

1. **Statement:** \( m \angle AXC = m \angle 1 + m \angle 2 \)
   - **Reason:** **Angle Addition Postulate**. This postulate tells us that if we have a larger angle made up of two adjacent angles, then the measure of the larger angle is the sum of the measures of the smaller angles.

2. **Statement:** \( m \angle DYF = m \angle 3 + m \angle 4 \)
   - **Reason:** **Angle Addition Postulate**. The same reasoning applies here as with \( m \angle AXC \).

3. **Statement:** \( m \angle AXC = m \angle DYF \)
   - **Reason:** **Given**. This equality was provided in the problem statement.

4. **Statement:** \( m \angle 1 + m \angle 2 = m \angle 3 + m \angle 4 \)
   - **Reason:** **Substitution Property**. Since \( m \angle AXC = m \angle DYF \), we can substitute the expressions from steps 1 and 2.

5. **Statement:** \( m \angle 2 = m \angle 4 \)
   - **Reason:** **Given**. This equality was also provided in the problem statement.

6. **Statement:** \( m \angle 1 + m \angle 4 = m \angle 3 + m \angle 4 \)
   - **Reason:** **Substitution Property**. Since \( m \angle 2 = m \angle 4 \), we can replace \( m \angle 2 \) in the equation from step 4 with \( m \angle 4 \).

7. **Statement:** \( m \angle 1 = m \angle 3 \)
   - **Reason:** **Subtraction Property of Equality**. By subtracting \( m \angle 4 \) from both sides of the equation in step 6, we get \( m \angle 1 = m \angle 3 \).

### **Summary of Completed Proof:**

| Statement                                  | Reason                              |
|--------------------------------------------|-------------------------------------|
| 1. \( m \angle AXC = m \angle 1 + m \angle 2 \)   | Angle Addition Postulate           |
| 2. \( m \angle DYF = m \angle 3 + m \angle 4 \)   | Angle Addition Postulate           |
| 3. \( m \angle AXC = m \angle DYF \)              | Given                              |
| 4. \( m \angle 1 + m \angle 2 = m \angle 3 + m \angle 4 \) | Substitution Property           |
| 5. \( m \angle 2 = m \angle 4 \)                  | Given                              |
| 6. \( m \angle 1 + m \angle 4 = m \angle 3 + m \angle 4 \) | Substitution Property           |
| 7. \( m \angle 1 = m \angle 3 \)                  | Subtraction Property of Equality   |

This completes the proof that \( m \angle 1 = m \angle 3 \).

Would you like further clarification on any of these steps?

### **Related Questions:**
1. How can we use the Angle Addition Postulate to solve other geometry problems?
2. What is the Substitution Property, and how is it useful in proofs?
3. Can you explain the Subtraction Property of Equality with another example?
4. What other properties can be used in geometric proofs similar to this one?
5. How do we identify which angles are congruent when analyzing different geometric figures?

### **Tip:**
When working on geometry proofs, always label and annotate diagrams clearly to visualize relationships between different angles or lines more effectively. This can make identifying congruences and applying properties much easier.

Prove that m∠1 = m∠3 using the given information: m∠AXC = m∠DYF and m∠2 = m∠4.

This geometry problem demonstrates how to prove that m∠1 = m∠3 using the given angle relationships. It involves the Angle Addition Postulate, Substitution Property, and Subtraction Property of Equality. Suitable for students in Grades 8-10.

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