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To prove the Consecutive Interior Angles Converse, we'll follow the steps and fill in the reasons in the provided table. 

Here is the setup from the image:

1. \(\angle 4\) and \(\angle 6\) are supplements (Given)
2. \(\angle 3\) and \(\angle 4\) are supplements
3. \(\angle 3 \cong \(\angle 6\)
4. \(l_1 \parallel l_2\)

Let's analyze each step and fill in the appropriate reasons:

1. \(\angle 4\) and \(\angle 6\) are supplements. 
   - **Reason:** Given

2. \(\angle 3\) and \(\angle 4\) are supplements. 
   - **Reason:** If two angles are supplements of the same angle (or congruent angles), then they are congruent (Supplementary Angles Theorem).

3. \(\angle 3 \cong \(\angle 6\)
   - **Reason:** If two angles are supplements of the same angle, they are congruent (by the definition of supplementary angles).

4. \(l_1 \parallel l_2\)
   - **Reason:** If two lines are cut by a transversal such that a pair of consecutive interior angles are supplementary, then the lines are parallel (Converse of the Consecutive Interior Angles Theorem).

Here is the completed table with reasons:

| Steps | Reasons |
|-------|---------|
| 1. \(\angle 4\) and \(\angle 6\) are supplements | Given |
| 2. \(\angle 3\) and \(\angle 4\) are supplements | If two angles are supplements of the same angle (or congruent angles), then they are congruent (Supplementary Angles Theorem) |
| 3. \(\angle 3 \cong \(\angle 6\) | If two angles are supplements of the same angle, they are congruent (by the definition of supplementary angles) |
| 4. \(l_1 \parallel l_2\) | If two lines are cut by a transversal such that a pair of consecutive interior angles are supplementary, then the lines are parallel (Converse of the Consecutive Interior Angles Theorem) |

Do you want more details or have any questions?

Here are 5 related questions that expand on this information:
1. What are consecutive interior angles in the context of parallel lines and a transversal?
2. How does the Converse of the Consecutive Interior Angles Theorem differ from the Consecutive Interior Angles Theorem?
3. Can you provide an example where consecutive interior angles are not supplementary? What does this indicate about the lines?
4. How do the properties of parallel lines help in solving geometric proofs involving angles?
5. What other angle relationships can be used to prove lines are parallel?

**Tip:** When working on geometric proofs, always remember to refer back to theorems and definitions you have learned. They are the building blocks for logical reasoning and proofs.

Prove the Consecutive Interior Angles Converse.

Learn how to prove the Consecutive Interior Angles Converse using parallel lines and a transversal. This geometric proof explains the relationships between angles and theorems used to determine if lines are parallel. Suitable for students in Grades 8-10.

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