This image shows a proof involving complementary angles and vertical angles. Let's solve the proof and fill in the "Reasons" column of the table.

Given:

• $\angle 1$ and $\angle 2$ are complements.
• Prove that $\angle 3$ and $\angle 4$ are complements.

Proof:

Statements:

1. $\angle 1$ and $\angle 2$ are complements.
2. $m\angle 1 + m\angle 2 = 90^\circ$
3. $\angle 1$ and $\angle 3$ are vertical angles, and $\angle 2$ and $\angle 4$ are vertical angles.
4. $m\angle 1 = m\angle 3$ and $m\angle 2 = m\angle 4$
5. $m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4$
6. $m\angle 3 + m\angle 4 = 90^\circ$
7. $\angle 3$ and $\angle 4$ are complements.

Reasons:

1. Given
2. Definition of complementary angles (If two angles are complements, their measures add up to $90^\circ$.)
3. Vertical angles theorem (Vertical angles are congruent.)
4. Congruent angles have equal measures (Vertical angles have the same measure.)
5. Substitution property (From Statements 2 and 4, substitute $m\angle 1$ with $m\angle 3$ and $m\angle 2$ with $m\angle 4$.)
6. Substitution property (Substitute $m\angle 1 + m\angle 2$ with $m\angle 3 + m\angle 4$ from Statement 5.)
7. Definition of complementary angles (If the sum of two angles is $90^\circ$, they are complements.)

Would you like further details or explanations on any part of the proof? Here are five related questions to expand your understanding:

1. How does the vertical angles theorem apply in geometry proofs?
2. What other angle relationships, besides complements, could be used in geometric proofs?
3. How can we distinguish between vertical angles and linear pairs in different diagrams?
4. What is the significance of substitution in proofs involving angle measures?
5. Could this proof be approached using a different method?

Tip: Always remember that vertical angles are congruent, which makes them key in solving many geometry problems involving angle relationships!