Question

the diagram below shows the construction of line m, parallel to line l, through point p. which theorem was used to justify this construction? 1) if two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. 2) if two lines are cut by a transversal and the interior angles on the same side are supplementary, the lines are parallel. 3) if two lines are perpendicular to the same line, they are parallel. 4) if two lines are cut by a transversal and the corresponding angles are congruent, they are parallel.

Explanation:

To determine the theorem justifying the construction of parallel line \( m \) through point \( P \) (parallel to \( l \)):

• The construction creates congruent corresponding angles (via arc - marking to replicate the angle from line \( l \) at point \( P \) for line \( m \)).
• Option 4 states: "If two lines are cut by a transversal and the corresponding angles are congruent, they are parallel", which matches the construction (corresponding angles are made congruent to ensure \( m \parallel l \)).
• Option 1 (alternate interior angles) and Option 2 (same - side interior angles) do not align with the construction’s method of replicating angles at the same relative position (corresponding angles). Option 3 (perpendicular to the same line) is irrelevant here as the lines are not perpendicular to a common line.

Answer:

1. If two lines are cut by a transversal and the corresponding angles are congruent, they are parallel.