Answer:
The next logical step in the proof is to apply the Corresponding Angles Converse Postulate, stating that if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
Step-by-step explanation:
Given that \( p \parallel q \) and \( r \parallel s \), we have corresponding angles that are congruent. Specifically, angle \( 1 \) corresponds to angle \( 14 \) because both are formed by the intersection of \( p \) and \( r \).
According to the Corresponding Angles Converse Postulate, if corresponding angles are congruent, then the lines are parallel.
Therefore, we can conclude that \( p \parallel r \). Since \( p \) is parallel to \( r \) and \( r \) is parallel to \( s \), by the Transitive Property of Parallel Lines, \( p \parallel s \).
Now, since \( p \parallel s \), angle \( 1 \) and angle \( 14 \) are corresponding angles of the parallel lines \( p \) and \( s \). Hence, angle \( 1 \) is congruent to angle \( 14 \).
We know that supplementary angles have a sum of \( 180^\circ \), so if angle \( 1 \) is congruent to angle \( 14 \), then angles \( 1 \) and \( 14 \) are supplementary.
Therefore, the next step is to explicitly state and use the Corresponding Angles Converse Postulate to establish that \( p \parallel s \), and then conclude that \( \angle 1 \) and \( \angle 14 \) are supplementary angles.