https://file.aiquickdraw.com/mathbot/file/1727179638656fx129g6h.jpg

Let's go step by step to complete the proof based on the information given in the image. 

### Problem:
- Given:
  1. \( \overline{AB} \perp \overline{BC} \)
  2. \( \angle 2 \) and \( \angle 3 \) are complementary.
  
- To prove: \( \angle 1 \cong \angle 2 \)

### Steps to Complete the Proof:

1. **Statement**: \( \overline{AB} \perp \overline{BC} \)
   - **Reason**: Given.
   
2. **Statement**: \( \angle ABC \) is a right angle.
   - **Reason**: Definition of perpendicular lines (since \( \overline{AB} \perp \overline{BC} \), \( \angle ABC \) is a right angle).

3. **Statement**: \( m\angle 1 + m\angle 3 = m\angle ABC \)
   - **Reason**: Definition of a right angle (since \( \angle ABC \) is a right angle, its measure is 90°, and \( \angle 1 \) and \( \angle 3 \) form this right angle).

4. **Statement**: \( m\angle 1 + m\angle 3 = 90^\circ \)
   - **Reason**: Substitution (from \( m\angle ABC = 90^\circ \)).

5. **Statement**: \( m\angle 2 + m\angle 3 = 90^\circ \)
   - **Reason**: Given (since \( \angle 2 \) and \( \angle 3 \) are complementary, their measures add up to 90°).

6. **Statement**: \( m\angle 1 = m\angle 2 \)
   - **Reason**: Transitive property (since both \( m\angle 1 + m\angle 3 = 90^\circ \) and \( m\angle 2 + m\angle 3 = 90^\circ \), by subtracting \( m\angle 3 \), you get \( m\angle 1 = m\angle 2 \)).

7. **Statement**: \( \angle 1 \cong \angle 2 \)
   - **Reason**: Definition of congruent angles (angles with equal measures are congruent).

This completes the proof that \( \angle 1 \cong \angle 2 \).

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Do you want further details on any of the steps, or would you like to discuss a particular part?

### Related Questions:
1. What is the transitive property of equality in geometry?
2. How do complementary angles work, and what are their properties?
3. How does the perpendicularity of lines help in identifying right angles?
4. Can you explain the difference between equal and congruent angles?
5. How would the proof change if angles 1 and 2 were not congruent?

### Tip:
Always double-check the given information in a proof, as it often provides the key reasons for each logical step.

Complete the following proof: Given AB ⊥ BC, ∠2 and ∠3 are complementary. Prove ∠1 ≅ ∠2.

Learn how to complete the proof for ∠1 ≅ ∠2 in a right triangle with complementary angles. This proof involves key concepts like perpendicular lines, complementary angles, and the transitive property. Suitable for students in Grades 8-10.

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