Consider the cuboid 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻, where 𝐴𝐵 is not equal to 𝐵𝐶 is not equal to 𝐴𝐸. What can be said about the line segment 𝐸𝐹 and the line segment 𝐹𝐺? What can be said about the line segment 𝐴𝐸 and the line segment 𝐶𝐺? What can be said about the line segment 𝐵𝐷 and the line segment 𝐷𝐻? What can be said about the line segment 𝐵𝐷 and the line segment 𝐴𝐶?
In this question, we are looking at the relationship between two line segments in a three-dimensional shape. So let’s consider the possible relationships that two lines in space can have. These are, namely, they can be parallel, perpendicular, neither parallel nor perpendicular, or skew.
Let’s consider what each of these four different relationships would actually look like. Firstly, let’s think about what happens if the two lines are actually on the same plane. Well, we could then have two parallel lines. Or if the lines intersect at 90 degrees, then they would be perpendicular. Or of course these two lines on the same plane might intersect at an angle which is not 90 degrees. And so we can say that they would be neither parallel nor perpendicular.
So this leaves us with the final type of relationship, which is two skew lines. The important thing to remember about skew lines is that they can only exist in three dimensions. They also do not intersect. So let’s say, for example, we have a line on a plane. Then, another line which is skew to the first line might look like this. So, as we go through each part of this question, we’ll look at the different line segments and consider the relationship between them. The fact that this is a cuboid will be very important in helping us determine the relationship between these line segments.
Part one asks, what can be said about the line segment 𝐸𝐹 and the line segment 𝐹𝐺? These two line segments are highlighted on the cuboid. Well, as this is a cuboid, we know that each face will be a rectangle. And the face will be on the same plane. We also know that rectangles have four 90-degree angles. Therefore, we can give the answer in a statement form about what can be said about the two given line segments is that they are perpendicular.
In the second part of this question, we are looking at the line segments 𝐴𝐸 and 𝐶𝐺. These two line segments lie on opposite faces of the rectangular prism. We know that they don’t intersect, and we know that they aren’t perpendicular. In fact, these two line segments are parallel. It might be confusing at this point since we know that they are on opposite faces. Well, why do we not say that these two lines are skew? And the answer is that, in fact, we can say that these four points all lie on the same plane, a plane which passes through the points 𝐴, 𝐶, 𝐺, and 𝐸. Since these two line segments lie on the same plane and they are parallel, then we give the answer that they are parallel.
The third part of this question asks, what can be said about the line segment 𝐵𝐷 and the line segment 𝐷𝐻? We know that line segment 𝐵𝐷 and line segment 𝐷𝐻 are line segments that occur on the perpendicular faces of the prism. And they intersect at point 𝐷. Therefore, line segment 𝐵𝐷 and line segment 𝐷𝐻 must be perpendicular.
In the final part of this question, we are asked about the line segment 𝐵𝐷 and the line segment 𝐴𝐶. Note that the line segment 𝐴𝐶 wasn’t on the original diagram, but we can add it in. Of course, we can see that these two line segments would actually form the face diagonals of this cuboid. We can also say then that these two line segments lie on the same plane. We know that these two line segments won’t be parallel, and the only way they would be perpendicular is if the face was a square. But since we are given in the question that side 𝐴𝐵 is not equal to 𝐵𝐶, then we know that this is a rectangle.
As the two line segments lie on the same plane, they will not be skew. Therefore, we can give the answer to the final part of this question that they are neither parallel nor perpendicular.