https://file.aiquickdraw.com/mathbot/file/1726712570055i0xat8hw.jpg

Let's go through the proof presented in the image step by step to complete the missing parts.

The problem asks to prove that ∠2 and ∠6 are complementary, given that ∠4 is a right angle and ∠1 is congruent to ∠2.

Here is the structure of the proof:

1. **Statement**: ∠4 is a right angle  
   **Reason**: Given

2. **Statement**: ∠1 ≅ ∠2  
   **Reason**: Given

3. **Statement**: m∠4 = 90°  
   **Reason**: Definition of right angle (from step 1)

4. **Statement**: m∠1 = m∠2  
   **Reason**: Definition of congruent angles (from step 2)

5. **Statement**: m∠1 + m∠6 = m∠4  
   **Reason**: Vertical Angle Theorem (the figure shows vertical angles formed, implying ∠1 + ∠6 equals ∠4)

6. **Statement**: m∠2 + m∠6 = 90°  
   **Reason**: Substitution (from steps 3 and 5)

7. **Statement**: m∠2 + m∠6 = 90°  
   **Reason**: Substitution (from steps 4 and 6)

8. **Statement**: ∠2 and ∠6 are complementary  
   **Reason**: Definition of complementary angles (angles whose sum is 90° from step 7)

This completes the proof!

Do you have any questions or want further details on any of the steps?

### 5 Related Questions:
1. What is the Vertical Angle Theorem, and why is it used here?
2. Why does the definition of congruent angles apply in step 4?
3. How can substitution be applied in mathematical proofs?
4. What are complementary angles, and why do they sum to 90°?
5. What are the differences between complementary and supplementary angles?

### Tip: 
When proving angles are complementary, always ensure you use the correct definition and check if their measures sum up to 90°.

Prove that ∠2 and ∠6 are complementary, given that ∠4 is a right angle and ∠1 ≅ ∠2.

This problem involves proving that ∠2 and ∠6 are complementary using angle congruence and the Vertical Angle Theorem. It starts with the given that ∠4 is a right angle and ∠1 ≅ ∠2, then uses substitution to complete the proof. Suitable for students in Grades 9-10.

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