Question

multiple answer 10 points given lines a and b are parallel, which of the following pairs of angles are supplementary? select all. <6 and <4 <6 and <7 <5 and <4 <1 and <2 <1 and <8 <3 and <5

Explanation:

Step1: Recall the properties of parallel lines and transversals

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Also, linear - pair angles are supplementary.

Step2: Analyze each pair

• For \(\angle6\) and \(\angle4\):

Since \(a\parallel b\), \(\angle4\) and \(\angle6\) are consecutive interior angles. By the consecutive - interior - angles theorem, if \(a\parallel b\) and \(t\) (the transversal) cuts them, then \(\angle4+\angle6 = 180^{\circ}\).

• For \(\angle6\) and \(\angle7\):

\(\angle6\) and \(\angle7\) form a linear pair. By the definition of a linear pair, \(\angle6+\angle7=180^{\circ}\) (since they are adjacent angles on a straight line).

• For \(\angle5\) and \(\angle4\):

Since \(a\parallel b\), \(\angle4\) and \(\angle5\) are alternate interior angles. Alternate interior angles are equal (\(\angle4=\angle5\) when \(a\parallel b\)), and they are not supplementary (unless each is \(90^{\circ}\)).

• For \(\angle1\) and \(\angle2\):

\(\angle1\) and \(\angle2\) form a linear pair. \(\angle1 + \angle2=180^{\circ}\) (by the definition of a linear pair).

• For \(\angle1\) and \(\angle8\):

Since \(a\parallel b\), \(\angle1=\angle5\) (corresponding angles). And \(\angle5+\angle8 = 180^{\circ}\) (linear pair). So \(\angle1+\angle8=180^{\circ}\) (by substitution).

• For \(\angle3\) and \(\angle5\):

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

Answer:

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