To complete this proof, we need to fill in the missing reason in line 3.

Proof Steps with Completed Reasoning

1. Statement: ∠H and ∠I are complementary.
   Reason: Given.

2. Statement: ∠J and ∠H are complementary.
   Reason: Given.

3. Statement: $m∠H + m∠I = 90^\circ$
   Reason: Definition of complementary angles. (This is the missing reason, as complementary angles add up to $90^\circ$).

4. Statement: $m∠J + m∠H = 90^\circ$
   Reason: Definition of complementary angles.

5. Statement: $m∠H + m∠I = m∠J + m∠H$
   Reason: Transitive Property of Equality (since both equal 90°).

6. Statement: $m∠I = m∠J$
   Reason: Properties of addition, subtraction, multiplication, and division.

7. Statement: ∠I ≅ ∠J
   Reason: Definition of congruence.

This completes the proof.

Would you like more details on any of the steps or concepts used here?

Additional Questions for Deeper Understanding

1. What is the difference between complementary and supplementary angles?
2. How does the transitive property apply to geometric proofs?
3. Why is the definition of congruence used in the last step of the proof?
4. Can complementary angles be congruent?
5. How would this proof change if the angles were supplementary?

Tip: When proving geometric relationships, always check if properties like equality, transitivity, or definitions (e.g., of complementary or supplementary angles) are applicable to simplify your reasoning.