Use the given information to prove that m<1=m<3 Given:m<CWE=m<FZH m<2=m<4 Prove:m<1=m<3  What is the third reason?

To prove that \( m \angle 1 = m \angle 3 \), let's break down the given information and proceed step by step.

### Given Information:
1. \( m \angle CWE = m \angle FZH \)  
   (This states that these two angles are congruent.)
2. \( m \angle 2 = m \angle 4 \)  
   (This states that these two angles are also congruent.)

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

### Proof:

1. **Given**: \( m \angle CWE = m \angle FZH \)
   - The first given information tells us these two angles are congruent.
   
2. **Given**: \( m \angle 2 = m \angle 4 \)
   - The second given states that these angles are congruent.

3. **Vertical Angles Theorem**:  
   - \( m \angle CWE = m \angle 1 \) and \( m \angle FZH = m \angle 3 \), by the **vertical angles theorem**. Vertical angles are always congruent.

4. **Transitive Property of Equality**:  
   - Since \( m \angle CWE = m \angle FZH \), and \( m \angle CWE = m \angle 1 \) and \( m \angle FZH = m \angle 3 \), we can conclude by the transitive property that:  
   \[
   m \angle 1 = m \angle 3
   \]

### Third Reason:
The third reason in the proof is the **vertical angles theorem**, which allows us to conclude that the angles are congruent based on the geometric property of vertical angles.

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Would you like further details or have any other questions?

### Related Questions:
1. What is the transitive property of equality in geometry?
2. Can vertical angles ever be unequal?
3. How does the vertical angles theorem work in real-world applications?
4. What other theorems are often used alongside the vertical angles theorem?
5. How would this proof change if the angles were supplementary instead of vertical?

**Tip:** Always consider the transitive property when dealing with congruent angles across different pairs of angles!

Learn how to prove that angles 1 and 3 are congruent using the Vertical Angles Theorem and the Transitive Property of Equality. Understand the principles of congruence in geometry with step-by-step explanations. Suitable for high school students studying geometry.

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