Question

given that lines a and b are parallel, which of the following pairs of angles are supplementary? select all that are correct. <1 and <2 <3 and <5 <6 and <4 <5 and <4 <1 and <7 <6 and <7

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles are two angles whose sum is \(180^{\circ}\).

Step2: Analyze \(\angle1\) and \(\angle2\)

\(\angle1\) and \(\angle2\) form a linear - pair. By the linear - pair postulate, \(\angle1+\angle2 = 180^{\circ}\).

Step3: Analyze \(\angle3\) and \(\angle5\)

Since lines \(a\) and \(b\) are parallel, \(\angle3\) and \(\angle5\) are same - side interior angles. By the same - side interior angles theorem, \(\angle3+\angle5=180^{\circ}\).

Step4: Analyze \(\angle6\) and \(\angle4\)

\(\angle6\) and \(\angle4\) are not supplementary. \(\angle4\) and \(\angle6\) are not in a relationship (such as linear - pair or same - side interior angles) that would make their sum \(180^{\circ}\).

Step5: Analyze \(\angle5\) and \(\angle4\)

\(\angle5\) and \(\angle4\) are not supplementary. There is no geometric relationship (linear - pair, same - side interior angles) between them to make their sum \(180^{\circ}\).

Step6: Analyze \(\angle1\) and \(\angle7\)

\(\angle1\cong\angle3\) (vertical angles). \(\angle3\) and \(\angle5\) are same - side interior angles (\(\angle3+\angle5 = 180^{\circ}\)), and \(\angle5\cong\angle7\) (vertical angles). So \(\angle1+\angle7=\angle3+\angle5 = 180^{\circ}\).

Step7: Analyze \(\angle6\) and \(\angle7\)

\(\angle6\) and \(\angle7\) form a linear - pair. By the linear - pair postulate, \(\angle6+\angle7 = 180^{\circ}\). But the problem is about lines \(a\) and \(b\) being parallel. However, \(\angle6\) and \(\angle7\) are a linear - pair (not related to the parallel - line property for this problem's context of using parallel - line theorems for supplementary angles in the given figure). But if we consider all supplementary pairs:

• For \(\angle1\) and \(\angle2\): linear - pair (\(\angle1+\angle2 = 180^{\circ}\)).
• For \(\angle3\) and \(\angle5\): same - side interior angles (\(\angle3+\angle5 = 180^{\circ}\)).
• For \(\angle1\) and \(\angle7\): \(\angle1=\angle3\) (vertical angles), \(\angle3+\angle5 = 180^{\circ}\), \(\angle5=\angle7\) (vertical angles), so \(\angle1+\angle7 = 180^{\circ}\).

Answer:

\(\angle1\) and \(\angle2\), \(\angle3\) and \(\angle5\), \(\angle1\) and \(\angle7\)