Question

practice: lesson 2-2 proving lines parallel (part 2) valeria
use the figure for exercises 1 - 5. using the given information, which lines can you conclude are parallel? state the theorem or postulate that justifies each answer.

1. ∠1≅∠5
2. ∠2≅∠3
3. ∠6≅∠7
4. ∠1≅∠8
5. ∠5 + ∠8 = 180

use the figure for exercises 6 - 10. using the given information, which lines can you conclude are parallel? state the theorem or postulate that justifies each answer.

1. ∠1≅∠4
2. ∠6≅∠7
3. ∠2≅∠4
4. ∠4≅∠8
5. ∠6 + ∠8 = 180

Explanation:

Step1: Recall parallel - line theorems

Corresponding - angles postulate: If corresponding angles are congruent, then the lines are parallel. Alternate - interior angles theorem: If alternate - interior angles are congruent, then the lines are parallel. Alternate - exterior angles theorem: If alternate - exterior angles are congruent, then the lines are parallel. Same - side interior angles theorem: If same - side interior angles are supplementary, then the lines are parallel.

Step2: Analyze exercise 1 - 5

1. $\angle1$ and $\angle5$ are not corresponding, alternate - interior, alternate - exterior or same - side interior angles related to any two lines in the figure, so no lines can be concluded to be parallel.
2. $\angle2$ and $\angle3$ are alternate - interior angles. So $d\parallel e$ by the alternate - interior angles theorem.
3. $\angle6$ and $\angle7$ are not corresponding, alternate - interior, alternate - exterior or same - side interior angles related to any two lines in the figure, so no lines can be concluded to be parallel.
4. $\angle1$ and $\angle8$ are alternate - exterior angles. So $s\parallel t$ by the alternate - exterior angles theorem.
5. $\angle5$ and $\angle8$ are same - side interior angles. So $d\parallel e$ by the same - side interior angles theorem.

Step3: Analyze exercise 6 - 10

1. $\angle1$ and $\angle4$ are alternate - exterior angles. So $r\parallel s$ by the alternate - exterior angles theorem.
2. $\angle6$ and $\angle7$ are not corresponding, alternate - interior, alternate - exterior or same - side interior angles related to any two lines in the figure, so no lines can be concluded to be parallel.
3. $\angle2$ and $\angle4$ are corresponding angles. So $p\parallel q$ by the corresponding - angles postulate.
4. $\angle4$ and $\angle8$ are corresponding angles. So $r\parallel s$ by the corresponding - angles postulate.
5. $\angle6$ and $\angle8$ are same - side interior angles. So $p\parallel q$ by the same - side interior angles theorem.

Answer:

1. No lines are parallel.
2. $d\parallel e$, alternate - interior angles theorem.
3. No lines are parallel.
4. $s\parallel t$, alternate - exterior angles theorem.
5. $d\parallel e$, same - side interior angles theorem.
6. $r\parallel s$, alternate - exterior angles theorem.
7. No lines are parallel.
8. $p\parallel q$, corresponding - angles postulate.
9. $r\parallel s$, corresponding - angles postulate.
10. $p\parallel q$, same - side interior angles theorem.