Question

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

1. $\angle 3\cong\angle 7$
2. $\angle 9\cong\angle 11$
3. $\angle 2\cong\angle 16$
4. $m\angle 5 + m\angle 12 = 180^{circ}$

Explanation:

1.

Step1: Analyze the relationship between ∠3 and ∠7

∠3 and ∠7 are corresponding angles. If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.

Step2: Determine the parallel lines

Since ∠3 ≅ ∠7, by the Corresponding Angles Theorem, line \(a\parallel b\).

2.

Step1: Analyze the relationship between ∠9 and ∠11

∠9 and ∠11 are alternate interior angles. If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

Step2: Determine the parallel lines

Since ∠9 ≅ ∠11, by the Alternate Interior Angles Theorem, line \(l\parallel m\).

3.

Step1: Analyze the relationship between ∠2 and ∠16

∠2 and ∠16 are not in a direct relationship (corresponding, alternate - interior, alternate - exterior, same - side interior) that would imply parallel lines for the lines in the figure.

Step2: Conclusion

There is no pair of parallel lines based on ∠2 ≅ ∠16.

4.

Step1: Analyze the relationship between ∠5 and ∠12

∠5 and ∠12 are same - side interior angles. If \(m\angle5 + m\angle12=180^{\circ}\), then by the Same - Side Interior Angles Theorem, if two lines are cut by a transversal and the sum of the same - side interior angles is \(180^{\circ}\), the lines are parallel.

Step2: Determine the parallel lines

Since \(m\angle5 + m\angle12 = 180^{\circ}\), line \(l\parallel m\).

Answer:

1. \(a\parallel b\) by the Corresponding Angles Theorem.
2. \(l\parallel m\) by the Alternate Interior Angles Theorem.
3. No parallel lines.
4. \(l\parallel m\) by the Same - Side Interior Angles Theorem.