Question

Consider transversal t cutting lines p and q.

Transversal t crosses lines p and q to form 8 angles. Clockwise from top left, the angles are 1, 2, 4, 3; 5, 6, 8, 7.

Which statement allows Carmen to conclude that line p and line q are parallel?
∠2 ≅ ∠6
∠1≅ ∠4
m∠1 + m∠2 = 180°
m∠5 + m∠7 = 180°

Answer 1

To determine which statement allows Carmen to conclude that lines \( p \) and \( q \) are parallel, we can analyze the relationships between the angles formed by transversal \( t \) intersecting lines \( p \) and \( q \).

1. **Corresponding Angles**: Angles that are in the same position relative to the lines and the transversal (e.g., ∠2 and ∠6) are congruent if the lines are parallel.

2. **Alternate Interior Angles**: Angles that lie between the two lines and on opposite sides of the transversal (e.g., ∠4 and ∠6) are congruent if the lines are parallel.

3. **Same Side Interior Angles**: Angles that lie between the two lines and on the same side of the transversal (e.g., ∠5 and ∠7) are supplementary if the lines are parallel.

### Evaluating the Statements:
- **∠2 ≅ ∠6**: This statement suggests that corresponding angles are equal; if this is true, then lines \( p \) and \( q \) must be parallel.
- **∠1 ≅ ∠4**: This is also a case of corresponding angles, so if true, it indicates the lines are parallel.
- **m∠1 + m∠2 = 180°**: This means that angles 1 and 2 are supplementary—which would suggest that lines are not parallel.
- **m∠5 + m∠7 = 180°**: This means that angles 5 and 7 are also supplementary, which is another indicator that the lines are not parallel.

### Conclusion:
Among the statements, the correct one that allows Carmen to conclude that \( p \) and \( q \) are parallel is:

**∠2 ≅ ∠6**.

This is because if corresponding angles are equal, then the lines cut by a transversal are confirmed to be parallel.