The image shows a flow proof diagram where we are asked to complete the reasoning for proving that lines $a \parallel b$ (parallel lines) given that $\angle 1$ and $\angle 3$ are supplementary. The structure of the flow proof is as follows:

Given:

• $\angle 1$ and $\angle 3$ are supplementary.

Prove:

• $a \parallel b$

Flow Proof Diagram Steps:

1. $\angle 1$ and $\angle 3$ are supplementary.

   • This is the Given part.

2. a. ?

   • This step follows from the given statement. Here, we conclude that $\angle 1$ and $\angle 2$ are a linear pair (using the definition of a linear pair).

3. b. ?

   • The justification here is that $\angle 1$ and $\angle 2$ are supplementary by the definition of a linear pair.

4. $\angle 1$ and $\angle 2$ are supplementary.

5. c. ?

   • This step asks for a reason that would lead to the next conclusion. The answer is the Transitive Property of Supplementary Angles, which states that if two angles are supplementary to the same angle, they are congruent.

6. Supplements of the same angle are congruent.

7. d. ?

   • Finally, from this, we conclude that $a \parallel b$ based on the Converse of the Corresponding Angles Postulate (if two lines and a transversal form congruent corresponding angles, the lines are parallel).

Completed Flow Proof:

1. $\angle 1$ and $\angle 3$ are supplementary. (Given)
2. $\angle 1$ and $\angle 2$ form a linear pair. (a: Definition of a linear pair)
3. $\angle 1$ and $\angle 2$ are supplementary. (b: Definition of supplementary angles)
4. $\angle 1$ and $\angle 2$ are supplementary.
5. $\angle 2$ and $\angle 3$ are congruent. (c: Supplements of the same angle are congruent)
6. $a \parallel b$ (d: Converse of the Corresponding Angles Postulate)

Would you like to go into any more details about specific steps or the reasoning process? Here are 5 related questions to further deepen your understanding:

1. What are the properties of supplementary angles, and how are they used in proofs?
2. How does the definition of a linear pair relate to proving angles are supplementary?
3. What is the Converse of the Corresponding Angles Postulate, and when is it used?
4. What is the transitive property, and how does it apply to angle relationships?
5. How do parallel lines influence the relationships between angles formed by a transversal?

Tip: Always remember that the converse of a geometric theorem is an important tool in proofs, often reversing the direction of the reasoning process.