Question

given ( m parallel n ), ( angle 1 cong angle 3 )
prove ( p parallel q )
complete the two - column proof

Explanation:

Step1: Use the Alternate - Interior Angles Theorem

Since \(m\parallel n\), by the Alternate - Interior Angles Theorem, \(\angle1\cong\angle2\) (when two parallel lines are cut by a transversal, alternate - interior angles are congruent).

Step2: Apply the Substitution Property

We know that \(\angle1\cong\angle3\) (given) and \(\angle1\cong\angle2\) (from Step 1). By the Substitution Property of Congruence (if \(a = b\) and \(a = c\), then \(b = c\)), we can say \(\angle2\cong\angle3\).

Step3: Use the Converse of the Alternate - Interior Angles Theorem

If \(\angle2\cong\angle3\), then by the Converse of the Alternate - Interior Angles Theorem (if two lines are cut by a transversal such that alternate - interior angles are congruent, then the two lines are parallel), \(p\parallel q\).

Answer:

1. $\angle1\cong\angle2$
2. Substitution Property of Congruence
3. If alternate - interior angles are congruent, then the lines are parallel