1. The problem states that \(\triangle 1 \cong \angle 5\) and asks which postulate or theorem justifies that lines \(p\) and \(q\) are parallel. 2. When two lines are cut by a transversal, certain angle relationships indicate parallelism. 3. The key angle pairs are: alternate interior angles, corresponding angles, alternate exterior angles, and consecutive interior angles. 4. The problem states \(\angle 1 \cong \angle 5\). These angles are located on opposite sides of the transversal and outside the two lines, making them alternate exterior angles. 5. The Alternate Exterior Angles Theorem Converse states: If alternate exterior angles are congruent, then the lines are parallel. 6. Since \(\angle 1 \cong \angle 5\) are alternate exterior angles and congruent, by the Alternate Exterior Angles Theorem Converse, lines \(p\) and \(q\) are parallel. 7. Therefore, the correct justification is the Alternate Exterior Angles Theorem Converse.