To solve the given problem, let’s analyze and provide the steps:

Problem:

• Given: $\angle 1$ and $\angle 4$ are supplementary.
• Prove: $a \parallel b$.

Explanation:

1. Definition of supplementary angles: Supplementary angles sum up to $180^\circ$.

   $\angle 1 + \angle 4 = 180^\circ.$

2. Understanding the diagram: The given angles seem to involve transversals intersecting two lines. Since $\angle 1$ and $\angle 4$ are supplementary, they align with the property of angles formed when a transversal crosses parallel lines.

3. Corresponding property: If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, the lines are parallel (Converse of the Consecutive Interior Angles Theorem).

4. Conclusion: By the Converse of the Consecutive Interior Angles Theorem: $a \parallel b.$

Proof Structure:

1. Statement: $\angle 1$ and $\angle 4$ are supplementary. Reason: Given.

2. Statement: $\angle 1 + \angle 4 = 180^\circ$. Reason: Definition of supplementary angles.

3. Statement: Lines $a$ and $b$ are parallel. Reason: Converse of the Consecutive Interior Angles Theorem.

Would you like further clarification or details on the theorem?

Expanding Questions:

1. What are the other methods to prove that two lines are parallel?
2. Can you explain the role of the transversal in determining parallel lines?
3. What is the significance of the Converse of the Consecutive Interior Angles Theorem in geometry?
4. How would this proof differ if the lines were not parallel?
5. Can you identify other angle relationships formed by a transversal?

Tip:

When working with geometry proofs, always identify the key properties of angles formed by a transversal to establish relationships effectively.