https://file.aiquickdraw.com/mathbot/file/17272858185072iljfta8.jpg

Let's solve this geometric proof step by step, based on the given information and diagram.

### Given:
1. \( m \angle CXE = m \angle FZH \)  (Angles are congruent)
2. \( m \angle 2 = m \angle 4 \)  (These angles are congruent)

### To Prove:
\( m \angle 1 = m \angle 3 \)

### Step-by-Step Breakdown:

| **Statement**                                   | **Reason**                                 |
|-------------------------------------------------|--------------------------------------------|
| 1. \( m \angle CXE = m \angle 1 + m \angle 2 \) | **Angle Addition Property**                |
| 2. \( m \angle FZH = m \angle 3 + m \angle 4 \) | **Angle Addition Property**                |
| 3. \( m \angle CXE = m \angle FZH \)            | **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**                   |

### Explanation:
1. Use the **Angle Addition Property** to express \( m \angle CXE \) and \( m \angle FZH \) as the sums of two angles each (steps 1 and 2).
2. Substitute the given information \( m \angle CXE = m \angle FZH \) (step 3) and apply it to the sums (step 4).
3. Since \( m \angle 2 = m \angle 4 \) is given, substitute this into the equation (step 5).
4. Simplify and subtract the common terms from both sides, leaving \( m \angle 1 = m \angle 3 \) (steps 6 and 7).

Would you like more details or have questions about this proof?

### Related Questions:
1. What is the **Angle Addition Property**, and how is it used in proofs?
2. How do you apply the **Substitution Property** in geometric proofs?
3. Why does the **Subtraction Property** work in step 7 of this proof?
4. Can you explain the importance of congruent angles in solving geometry problems?
5. How would the proof change if \( m \angle 2 \neq m \angle 4 \)?

#### Tip:
When working on geometric proofs, always label your diagrams clearly and keep track of known relationships between angles and lines.

Given m∠CXE = m∠FZH, m∠2 = m∠4, prove m∠1 = m∠3. Complete the proof with the correct reasons.

Solve a geometric proof using the Angle Addition Property and Substitution. Given m∠CXE = m∠FZH and m∠2 = m∠4, learn how to prove that m∠1 = m∠3 step-by-step, applying important geometric theorems like the Angle Addition Property.

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