Question Video: Identifying a Correct Statement Using Geometric Properties Mathematics

Video Transcript

If 𝐹 is a point on the line between 𝐴 and 𝐵 and 𝐸 is a point on the line between 𝐶 and 𝐷, the line between 𝐴 and 𝐵 is perpendicular to the line between 𝐸 and 𝐹, and the line between 𝐶 and 𝐷 is perpendicular to the line between 𝐸 and 𝐹, which of the following is correct? Option (A) the line between 𝐴 and 𝐵 is parallel to the line between 𝐶 and 𝐷. Option (B) line segment 𝐸𝐹 bisects line segment 𝐴𝐵. Option (C) the line between 𝐴 and 𝐵 is parallel to the line between 𝐸 and 𝐹. Option (D) the line between 𝐴 and 𝐵 is perpendicular to the line between 𝐶 and 𝐷. Or is it option (E) line segment 𝐸𝐹 bisects line segment 𝐶𝐷?

In this question, we are given some information about some lines and asked to determine which of five given options must be true. To do this, let’s start by sketching the given information, that is, that the line between 𝐴 and 𝐵 is perpendicular to the line between 𝐸 and 𝐹 and that the line between 𝐶 and 𝐷 is also perpendicular to this line, where 𝐸 and 𝐹 lie on the lines as shown. In our sketch, we see that the line between 𝐸 and 𝐹 is a transversal of the other two lines. And we see that the corresponding angles of this transversal both have equal measure since they are both right angles.

This is enough to confirm that the line between 𝐴 and 𝐵 and the line between 𝐶 and 𝐷 must be parallel since the corresponding angles of the transversal have equal measure. Alternatively, we can use the specific result that any two lines perpendicular to the same line are parallel. Hence, the answer is option (A): the line between 𝐴 and 𝐵 is parallel to the line between 𝐶 and 𝐷.

It is worth noting that we can show that none of the other options must be true using our sketch. First, we can choose points 𝐸 and 𝐹 anywhere on the lines. So, we can choose point 𝐹 to not be the midpoint of line segment 𝐴𝐵. Therefore, option (B) is not true in general. The same reasoning works to show that option (E) is not true in general, since we can choose point 𝐸 to not be the midpoint of the line segment 𝐶𝐷. We are told that the line segment between 𝐴 and 𝐵 is perpendicular to the line between 𝐸 and 𝐹, so they are not parallel. Therefore, option (C) is not true.

Finally, since we have shown that the line between 𝐴 and 𝐵 is parallel to the line between 𝐶 and 𝐷, we can conclude that they do not intersect and so they are not perpendicular. Hence, the only statement that is true in general is option (A).