Question

1. given: ∠3≅∠5
2. given: ∠4≅∠8
3. given: ∠1≅∠7
4. given: ∠1 is supplementary to ∠8
5. given: ∠4 is supplementary to ∠6

name the postulate or theorem that can be used to prove the lines parallel.

Explanation:

Step1: Recall parallel - line postulates and theorems

For two lines cut by a transversal, corresponding - angles postulate states that if corresponding angles are congruent, the lines are parallel; alternate - interior angles theorem states that if alternate - interior angles are congruent, the lines are parallel; same - side interior angles theorem states that if same - side interior angles are supplementary, the lines are parallel.

Step2: Analyze each case

1. Given $\angle3\cong\angle5$, by the alternate - interior angles theorem, the lines are parallel.
2. Given $\angle4\cong\angle8$, by the corresponding angles postulate, the lines are parallel.
3. Given $\angle1\cong\angle7$, by the alternate - exterior angles theorem, the lines are parallel.
4. Given $\angle1$ is supplementary to $\angle8$, since $\angle1$ and $\angle8$ are same - side exterior angles, by the same - side exterior angles theorem (which is similar to same - side interior angles theorem), the lines are parallel.
5. Given $\angle4$ is supplementary to $\angle6$, since $\angle4$ and $\angle6$ are same - side interior angles, by the same - side interior angles theorem, the lines are parallel.

Answer:

1. Alternate - interior angles theorem
2. Corresponding angles postulate
3. Alternate - exterior angles theorem
4. Same - side exterior angles theorem
5. Same - side interior angles theorem