Question Video: Identifying the True Statement Relating to Given Parallel Lines and Traversals Mathematics

Video Transcript

If line 𝐴𝐵 is parallel to line 𝐶𝐷 and line 𝐸𝐹 is perpendicular to line 𝐶𝐷, which of the following is correct? (A) Line segment 𝐸𝐺 bisects line segment 𝐴𝐵. (B) Line 𝐴𝐵 is parallel to line 𝐸𝐺. (C) Line 𝐴𝐵 is perpendicular to line 𝐶𝐷. (D) Line 𝐴𝐵 is perpendicular to line 𝐸𝐺. (E) Line segment 𝐸𝐹 bisects line segment 𝐶𝐷.

Let’s begin by adding the fact that line 𝐸𝐹 being perpendicular to line 𝐶𝐷 means the lines are at right angles and the fact that lines 𝐴𝐵 and 𝐶𝐷 are parallel to our diagram. We can then use corresponding angles to note that line 𝐸𝐹 is perpendicular to line 𝐴𝐵. This shows that line 𝐴𝐵 is perpendicular to line 𝐸𝐺, which is answer (D). However, this is a particular case of the fact that if a line is perpendicular to line 𝐿, then it is perpendicular to any line parallel to 𝐿.

It is worth noting that although the diagram looks like the transversal bisects the parallel lines, this does not need to be, since we can translate the transversal and still have the image being perpendicular to the parallel lines. We can see that line 𝐿 is perpendicular to the parallel lines 𝐴𝐵 and 𝐶𝐷 and that it is not a perpendicular bisector of either line. Thus, we cannot conclude that line segment 𝐸𝐺 bisects line segment 𝐴𝐵 or that it bisects line segment 𝐶𝐷. So answers (A) and (E) are incorrect. Hence, line 𝐴𝐵 is perpendicular to line 𝐸𝐺, and the correct answer is option (D).