This is a flow proof for proving that lines $a \parallel b$ given that $\angle 1$ and $\angle 3$ are supplementary. The proof uses properties of supplementary angles and the relationships between angles formed by transversals and parallel lines. Here's the step-by-step solution to complete the flow proof:

1. $\angle 1$ and $\angle 3$ are supplementary (Given)

• Reason (a): Given.

2. $\angle 1$ and $\angle 2$ form a linear pair.

• Reason (b): Definition of a linear pair.

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

• Reason (c): If two angles form a linear pair, they are supplementary.

4. $\angle 3$ and $\angle 2$ are congruent.

• Reason (d): Supplements of the same angle are congruent (since both $\angle 1$ and $\angle 3$ are supplementary to $\angle 2$).

5. $a \parallel b$

• Reason (e): If alternate interior angles are congruent, the lines are parallel (this is the alternate interior angles theorem).

This completes the proof that $a \parallel b$ based on the relationships between angles and their congruency.

Would you like further details on any step? Here are five related questions that expand on this proof:

1. What is the definition of supplementary angles?
2. How do linear pairs relate to supplementary angles?
3. What are alternate interior angles, and how do they relate to parallel lines?
4. Can the converse of the alternate interior angles theorem be used to prove lines are not parallel?
5. What is the significance of the transitive property in angle congruence?

Tip: Understanding different angle relationships formed by transversals is crucial for proving lines are parallel or perpendicular.