Question

given: (angle 1congangle 3)
prove: (pparallel q)
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 (square). we see that (square) is congruent to (square) by the alternate interior angles theorem. therefore, angle 1 is congruent to angle 2 by the transitive property. so, we can conclude that lines (p) and (q) are parallel by the (square)

Explanation:

Step1: Reason for \(l\parallel m\)

Given \( \angle1\cong\angle3\). By the converse of the alternate - interior angles theorem (if two lines are cut by a transversal and the alternate - interior angles are congruent, then the lines are parallel), \(l\parallel m\).

Step2: Angles for alternate - interior angles theorem

Since \(l\parallel m\) and we want to use the alternate - interior angles theorem for another pair of lines. The angles involved in the alternate - interior angles theorem for lines \(p\) and \(q\) (cut by transversal \(m\)) are \(\angle3\) and \(\angle2\). So \(\angle3\cong\angle2\) by the alternate - interior angles theorem (because \(l\parallel m\) and \(p\) and \(q\) are transversals in the context of the parallel lines \(l\) and \(m\)).

Step3: Theorem for \(p\parallel q\)

Since \(\angle1\cong\angle3\) (given) and \(\angle3\cong\angle2\) (alternate - interior angles theorem for \(l\parallel m\)), then \(\angle1\cong\angle2\) (transitive property). By the converse of the corresponding angles theorem (if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel), \(p\parallel q\).

Answer:

1. Given (or the converse of the alternate - interior angles theorem, since \(\angle1\) and \(\angle3\) are alternate - interior angles for lines \(l\) and \(m\) with an appropriate transversal).
2. \(\angle3\)
3. \(\angle2\)
4. converse of the corresponding angles theorem.