Question

determine which lines, if any, must be parallel. select all that apply. explain your reasoning. the lines are parallel by the linear pair perpendicular theorem. the lines are parallel by the perpendicular transversal theorem. the lines are parallel by the lines perpendicular to a transversal theorem. you need to know something about both of the intersections between the transversal and the two lines

Explanation:

Step1: Recall parallel - line theorems

If two lines are perpendicular to the same line, then they are parallel. In the given figure, lines \(a\) and \(b\) are both perpendicular to line \(c\), and lines \(c\) and \(d\) are both perpendicular to line \(a\) (and \(b\)).

Step2: Identify parallel lines

Lines \(a\) and \(b\) are parallel because they are both perpendicular to the trans - versal \(c\). Lines \(c\) and \(d\) are parallel because they are both perpendicular to the trans - versal \(a\) (or \(b\)). This is based on the Lines Perpendicular to a Transversal Theorem which states that in a plane, if two lines are perpendicular to the same line, then they are parallel.

Answer:

a and b
c and d
The lines are parallel by the Lines Perpendicular to a Transversal Theorem.