Question

practice
example 1
use the given information to determine which lines, if any, are
parallel. state the postulate or theorem that justifies your answer.

1. ∠3 ≅ ∠7
2. ∠9 ≅ ∠11
3. ∠2 ≅ ∠16
4. m∠5 + m∠12 = 180°

Explanation:

Step1: Recall the Alternate Interior Angles Theorem

If two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel.
For \(\angle3\cong\angle7\), lines \(a\) and \(b\) are cut by transversal \(\ell\). But \(\angle3\) and \(\angle7\) are not alternate - interior angles for any pair of lines.
For \(\angle9\cong\angle11\), lines \(a\) and \(b\) are cut by transversal \(m\). \(\angle9\) and \(\angle11\) are vertical angles, not related to parallel - line criteria.
For \(\angle2\cong\angle16\), \(\angle2\) and \(\angle16\) are not in a position (corresponding, alternate - interior, etc.) to imply parallel lines.
For \(m\angle5 + m\angle12=180^{\circ}\), \(\angle5\) and \(\angle12\) are same - side interior angles. If two lines are cut by a transversal and the sum of the same - side interior angles is \(180^{\circ}\), then the two lines are parallel. Here, lines \(\ell\) and \(m\) are cut by transversal \(b\).

Answer:

Lines \(\ell\) and \(m\) are parallel. The Same - Side Interior Angles Theorem justifies the answer.