Lesson Video: Parallel Lines and Transversals: Other Relationships Mathematics

Video Transcript

In this video, we will learn how to apply the angle relationships between a pair of parallel lines and a transversal to establish and use other relationships between parallel lines and transversals. There are many uses for the angle relationships between parallel lines and transversals. The most common use is finding the measures of other angles using results such as the alternate interior angles being congruent. However, there are other uses, such as proving relationships between transversals of lines.

To see this in action, we first recall that a transversal of a pair of parallel lines will have congruent corresponding angles. This is a useful result, since we can use this to determine the angles between two lines using the angle between one line and a parallel line. In particular, if we have a line 𝐴𝐵 that is perpendicular to another line 𝐸𝐹, then we know that any line parallel to 𝐴𝐵 must have a corresponding angle with a measure of 90 degrees. This gives us the following result. If a line is perpendicular to line 𝐴𝐵, then it is perpendicular to any line parallel to line 𝐴𝐵. This is not the only property we can show using properties of transversals.

We recall that a transversal of a pair of parallel lines will have congruent corresponding angles. We can also recall that if the corresponding angles in a transversal of two lines are congruent, then the lines are parallel. Therefore, if a transversal of two lines is perpendicular to the two lines, then the lines must be parallel, since they have congruent corresponding angles. This gives us the following property. If two lines are perpendicular to the same line, then they must be parallel. Let’s now see an example of applying these properties to determine relationships between lines.

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).

Before we move on to our next example, there is another useful property we can show. Let’s consider a pair of distinct parallel lines, 𝐴𝐵 and 𝐶𝐷, and another pair of distinct parallel lines, 𝐶𝐷 and 𝐸𝐹. It appears as though all three of the lines are parallel. We can prove this is the case by sketching a transversal perpendicular to line 𝐸𝐹. Using corresponding angles, we can show that all three lines are perpendicular to the transversal. Then, we can recall that if two lines are perpendicular to the same line, then they must be parallel to each other. Hence, all three lines are parallel.

We have proven the following result. If two distinct lines are parallel to a third distinct line, then all three lines are parallel. Let’s now see an example of using this property to determine the relationship between given lines.

Fill in the blank. If line 𝐴𝐵 is parallel to line 𝐶𝐷 and line 𝐴𝐵 is parallel to line 𝐸𝐹, then line 𝐶𝐷 is what to line 𝐸𝐹.

We first recall that if two lines are parallel to the same line, then they must be parallel. Since both lines 𝐶𝐷 and 𝐸𝐹 are parallel to the same line, 𝐴𝐵, we must have that the two lines are parallel. Hence, the answer is that line 𝐶𝐷 is parallel to line 𝐸𝐹, denoted as shown.

There is one final property that we want to show about the transversals of parallel lines. And we will introduce this property with an example.

Consider the following diagram, where 𝐿 one, 𝐿 two, and 𝐿 three are all parallel and the lengths of the line segments between the parallel lines are as shown. In the diagram, it appears that if the lengths of the line segments of a transversal between three parallel lines are equal, then the lengths of the line segments of any transversal between three parallel lines will be equal. We can prove this by using congruent triangles.

We can start with a perpendicular transversal that is split into two sections of equal length by the parallel lines as shown. We can note that the measure of angle 𝑀𝑁𝐿 is equal to the measure of angle 𝑂𝑁𝑃, as they are vertically opposite angles. We now see that triangle 𝑂𝑁𝑃 and triangle 𝑀𝑁𝐿 are congruent by the ASA criterion. In particular, this means that 𝐿𝑁 is equal to 𝑁𝑃. Hence, the other transversal is also split into two sections of equal length. We can always add in a transversal perpendicular to these lines. So the proof of this result in general is very similar.

We have shown the following property. If a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length. Let’s now see some examples of using this property to find the length of a transversal between parallel lines.

If lines 𝐴𝐵, 𝐶𝐷, and 𝐸𝐹 are all parallel and 𝐶𝐸 equals two centimeters, find 𝐴𝐸.

We can see in the diagram that we are given three parallel lines and two transversals. We can also see that 𝐵𝐷 is equal to 𝐷𝐹. So, in this case, the parallel lines divide one of the transversals into line segments of equal length. We can recall that if a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length. Therefore, it will divide the other transversal into sections of equal length. Thus, 𝐴𝐶 is equal to 𝐶𝐸, which is equal to two centimeters. We note that 𝐴𝐸 is equal to 𝐴𝐶 plus 𝐶𝐸. So 𝐴𝐸 is equal to two centimeters plus two centimeters, which equals four centimeters.

In our final example, we will apply multiple properties of the transversals of parallel lines to a triangle with a side bisected by parallel lines.

Consider triangle 𝐴𝐵𝐶 and lines 𝐴𝑀 and 𝐸𝐷, which are parallel to line 𝐶𝐵. Find the length of the line segment 𝐴𝐵. Find the measure of angle 𝐴𝐵𝐶.

We are given three parallel lines and two transversals of these lines. We can then recall that if a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length. Since 𝐴𝐸 is equal to 𝐸𝐶, the segments of the other transversal must be equal in length. So 𝐴𝐷 is equal to 𝐷𝐵, which equals five millimeters. Since 𝐴𝐵 is equal to 𝐴𝐷 plus 𝐷𝐵, then 𝐴𝐵 equals five millimeters plus five millimeters, which equals 10 millimeters. 𝐴𝐵 is equal to 10 millimeters.

It appears in the diagram that triangle 𝐴𝐵𝐶 is a right triangle. However, we need to justify why this is the case. We can do this by recalling that if a line is perpendicular to line 𝐿, then it is perpendicular to any line parallel to 𝐿. Since line 𝐸𝐷 is perpendicular to line 𝐴𝐶 and line 𝐸𝐷 is parallel to line 𝐵𝐶, we must have that lines 𝐵𝐶 and 𝐴𝐶 are perpendicular. This means the angle at 𝐶 has a measure of 90 degrees, so 𝐴𝐵𝐶 is a right triangle.

The sum of the measures of the interior angles in a triangle is 180 degrees. So 180 degrees equals 35 degrees plus 90 degrees plus the measure of angle 𝐴𝐵𝐶. Rearranging the equation, we have the measure of angle 𝐴𝐵𝐶 equals 180 degrees minus 35 degrees minus 90 degrees, which equals 55 degrees. The answers to the two parts of this question are 10 millimeters and 55 degrees.

Let’s finish by recapping some of the most important points from this video. If a line is perpendicular to line 𝐴𝐵, then it is perpendicular to any line parallel to line 𝐴𝐵. If two lines are perpendicular to the same line, then they must be parallel. If two distinct lines are parallel to a third distinct line, then all three lines are parallel. If a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length.