Question

supplementary angle pairs that go with parallel lines
use the picture to answer the following questions:
list four angle pairs that must be supplementary when given two lines a||b are cut by a transversal (do not use linear pairs)
∠1 and ∠2
∠1 and ∠3
∠3 and ∠5
∠1 and ∠7
∠4 and ∠6
∠2 and ∠8
∠3 and ∠4
∠5 and ∠6

Explanation:

Step1: Recall the property of same - side interior angles

When two parallel lines \(a\parallel b\) are cut by a transversal, same - side interior angles are supplementary.
For lines \(a\) and \(b\) cut by the transversal, \(\angle3\) and \(\angle5\) are same - side interior angles. So \(\angle3+\angle5 = 180^{\circ}\).

Step2: Recall the property of same - side exterior angles

Same - side exterior angles are also supplementary. \(\angle2\) and \(\angle8\) are same - side exterior angles. So \(\angle2+\angle8=180^{\circ}\).

Step3: Use the property of corresponding angles and linear pairs

Since \(a\parallel b\), \(\angle1=\angle5\) (corresponding angles). And \(\angle5+\angle7 = 180^{\circ}\) (linear pair). By substitution, \(\angle1+\angle7=180^{\circ}\).

Step4: Use the property of corresponding angles and linear pairs

Since \(a\parallel b\), \(\angle4=\angle6\) (corresponding angles). And \(\angle6+\angle8 = 180^{\circ}\) (linear pair). Also, \(\angle4+\angle6\) is not correct. But since \(a\parallel b\), \(\angle4\) and \(\angle6\). Wait, no. Let's re - check.
Since \(a\parallel b\), \(\angle3=\angle7\) (corresponding angles). \(\angle1+\angle3 = 180^{\circ}\) (linear pair). But we need non - linear pairs.
Since \(a\parallel b\), \(\angle4\) and \(\angle6\): \(\angle4\) and \(\angle6\) are same - side interior angles. So \(\angle4+\angle6 = 180^{\circ}\)

Answer:

\(\angle3\) and \(\angle5\), \(\angle1\) and \(\angle7\), \(\angle4\) and \(\angle6\), \(\angle2\) and \(\angle8\)