Question

complete the missing parts of the paragraph proof.we know that angle 1 is congruent to angle 3 and that line ( l ) is parallel to line ( m ) because of the transitive property. we see that is congruent to angle 2 by the alternate interior angles. therefore, angle 1 is congruent to angle 2 by the transitive. we can conclude that lines ( p ) and ( q ) are parallel by the

Explanation:

Step1: Alternate Interior Angles

When two parallel lines \( l\) and \( m\) are cut by a transversal, alternate interior angles are congruent. Here, if we assume the transversal is the line that forms the angles in question, and since \( l\parallel m\), angle \( 3\) (assuming it is one of the angles formed by the transversal with \( l\)) is congruent to angle \( 2\) (the alternate - interior angle with respect to the transversal and parallel lines \( l\) and \( m\)).

Step2: Corresponding Angles Converse

The converse of the corresponding angles theorem states that if two lines are cut by a transversal such that a pair of corresponding angles are congruent, then the two lines are parallel. We know \(\angle1\cong\angle2\). If \(\angle1\) and \(\angle2\) are corresponding angles (formed by lines \( p\) and \( q\) and a transversal), then by the converse of the corresponding angles theorem, \(p\parallel q\).

Answer:

The last blank should be filled with "converse of the corresponding angles theorem".