Geometry worksheet featuring parallel lines, transversals, and angle theorems for practice.

Now I need to analyze each row of the table to determine if the given angle relationships prove that lines are parallel, and if so, which theorem applies.

Let me go through each one:

1. m∠2 = m∠6
- Angles 2 and 6 are corresponding angles.
- If corresponding angles are equal, then the lines are parallel (Corresponding Angles Postulate).
- Transversal: line c
- Parallel lines: a || b
- Reason: A (Converse of Corresponding Angles Postulate)

2. m∠3 + m∠5 = 180°
- Angles 3 and 5 are consecutive interior angles.
- If consecutive interior angles are supplementary, then the lines are parallel (Consecutive Interior Angles Theorem).
- Transversal: line c
- Parallel lines: a || b
- Reason: D (Converse of Same Side Interior Angles Theorem)

3. m∠2 = m∠7
- Angles 2 and 7 are alternate exterior angles.
- If alternate exterior angles are equal, then the lines are parallel (Alternate Exterior Angles Theorem).
- Transversal: line c
- Parallel lines: a || b
- Reason: C (Converse of Alternate Exterior Angles Theorem)

4. m∠3 = m∠6
- Angles 3 and 6 are alternate interior angles.
- If alternate interior angles are equal, then the lines are parallel (Alternate Interior Angles Theorem).
- Transversal: line c
- Parallel lines: a || b
- Reason: B (Converse of Alternate Interior Angles Theorem)

5. m∠1 = m∠8
- Angles 1 and 8 are alternate exterior angles.
- If alternate exterior angles are equal, then the lines are parallel (Alternate Exterior Angles Theorem).
- Transversal: line c
- Parallel lines: a || b
- Reason: C (Converse of Alternate Exterior Angles Theorem)

6. m∠2 + m∠3 = 180°
- Angles 2 and 3 are adjacent angles on the same line, not a relationship that proves parallelism.
- This is not a valid condition for proving lines are parallel.
- Reason: — (not applicable)

7. m∠1 = m∠5
- Angles 1 and 5 are corresponding angles.
- If corresponding angles are equal, then the lines are parallel (Corresponding Angles Postulate).
- Transversal: line c
- Parallel lines: a || b
- Reason: A (Converse of Corresponding Angles Postulate)

8. m∠4 = m∠6
- Angles 4 and 6 are alternate interior angles.
- If alternate interior angles are equal, then the lines are parallel (Alternate Interior Angles Theorem).
- Transversal: line c
- Parallel lines: a || b
- Reason: B (Converse of Alternate Interior Angles Theorem)

9. m∠3 = m∠7
- Angles 3 and 7 are corresponding angles.
- If corresponding angles are equal, then the lines are parallel (Corresponding Angles Postulate).
- Transversal: line c
- Parallel lines: a || b
- Reason: A (Converse of Corresponding Angles Postulate)

10. m∠4 + m∠5 = 180°
- Angles 4 and 5 are consecutive interior angles.
- If consecutive interior angles are supplementary, then the lines are parallel (Consecutive Interior Angles Theorem).
- Transversal: line c
- Parallel lines: a || b
- Reason: D (Converse of Same Side Interior Angles Theorem)

11. m∠1 = m∠6
- Angles 1 and 6 are not a standard pair for proving parallelism (they are neither corresponding, alternate interior, alternate exterior, nor consecutive interior angles).
- This is not a valid condition for proving lines are parallel.
- Reason: — (not applicable)

12. m∠4 = m∠8
- Angles 4 and 8 are corresponding angles.
- If corresponding angles are equal, then the lines are parallel (Corresponding Angles Postulate).
- Transversal: line c
- Parallel lines: a || b
- Reason: A (Converse of Corresponding Angles Postulate)

13. m∠2 + m∠8 = 180°
- Angles 2 and 8 are not a standard pair for proving parallelism (they are neither corresponding, alternate interior, alternate exterior, nor consecutive interior angles).
- This is not a valid condition for proving lines are parallel.
- Reason: — (not applicable)

14. m∠1 + m∠4 = 180°
- Angles 1 and 4 are adjacent angles on the same line, not a relationship that proves parallelism.
- This is not a valid condition for proving lines are parallel.
- Reason: — (not applicable)

Based on my analysis, I have completed the worksheet by finding the values of x for the first four diagrams and determining whether lines are parallel in the second part, along with the appropriate reasons.