Question

use the figure and the given information to determine which lines, if any, are parallel. choose the correct justification, if necessary

given: ( mangle 5 + mangle 12 = 180^circ )

options:

• ( p parallel q ) with justifications like if corresponding angles are congruent, then lines are parallel, if alternate exterior angles are congruent, then lines are parallel
• ( m parallel n ) with justifications like if consecutive interior angles are supplementary, then lines are parallel, if alternate interior angles are congruent, then lines are parallel
• it cannot be determined if any of the lines are parallel.

Explanation:

Step1: Identify Angle Relationship

First, we analyze the given angles \( \angle 5 \) and \( \angle 12 \). We need to determine their positions relative to lines \( m \), \( n \), and transversal \( q \). \( \angle 5 \) is on line \( m \), \( \angle 12 \) is on line \( n \), and both are between the two lines (interior) and on the same side of transversal \( q \) (consecutive). So they are consecutive interior angles.

Step2: Apply Parallel Line Theorem

The theorem states that if consecutive interior angles are supplementary (their measures add up to \( 180^\circ \)), then the two lines cut by the transversal are parallel. Given \( m\angle 5 + m\angle 12 = 180^\circ \), by the "consecutive interior angles supplementary implies lines parallel" theorem, lines \( m \) and \( n \) are parallel. The justification is "If consecutive interior angles are supplementary, then lines are parallel."

Answer:

\( m \parallel n \) (justification: If consecutive interior angles are supplementary, then lines are parallel.)