Lesson Video: Parallel Lines and Transversals: Other Relationships | Nagwa Lesson Video: Parallel Lines and Transversals: Other Relationships | Nagwa

# Lesson Video: Parallel Lines and Transversals: Other Relationships Mathematics

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.

13:41

### 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 π΄π΅ that is perpendicular to another line πΈπΉ, then we know that any line parallel to π΄π΅ must have a corresponding angle with a measure of 90 degrees. This gives us the following result. If a line is perpendicular to line π΄π΅, then it is perpendicular to any line parallel to line π΄π΅. 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. Letβs now see an example of applying these properties to determine relationships between lines.

If line π΄π΅ is parallel to line πΆπ· and line πΈπΉ is perpendicular to line πΆπ·, which of the following is correct? (A) Line segment πΈπΊ bisects line segment π΄π΅. (B) Line π΄π΅ is parallel to line πΈπΊ. (C) Line π΄π΅ is perpendicular to line πΆπ·. (D) Line π΄π΅ is perpendicular to line πΈπΊ. (E) Line segment πΈπΉ bisects line segment πΆπ·.

Letβs begin by adding the fact that line πΈπΉ being perpendicular to line πΆπ· means the lines are at right angles and the fact that lines π΄π΅ and πΆπ· are parallel to our diagram. We can then use corresponding angles to note that line πΈπΉ is perpendicular to line π΄π΅. This shows that line π΄π΅ is perpendicular to line πΈπΊ, which is answer (D). However, this is a particular case of the fact that if a line is perpendicular to line πΏ, then it is perpendicular to any line parallel to πΏ.

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 πΏ is perpendicular to the parallel lines π΄π΅ and πΆπ· and that it is not a perpendicular bisector of either line. Thus, we cannot conclude that line segment πΈπΊ bisects line segment π΄π΅ or that it bisects line segment πΆπ·. So answers (A) and (E) are incorrect. Hence, line π΄π΅ is perpendicular to line πΈπΊ, and the correct answer is option (D).

Before we move on to our next example, there is another useful property we can show. Letβs consider a pair of distinct parallel lines, π΄π΅ and πΆπ·, and another pair of distinct parallel lines, πΆπ· and πΈπΉ. 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 πΈπΉ. 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. Letβs now see an example of using this property to determine the relationship between given lines.

Fill in the blank. If line π΄π΅ is parallel to line πΆπ· and line π΄π΅ is parallel to line πΈπΉ, then line πΆπ· is what to line πΈπΉ.

We first recall that if two lines are parallel to the same line, then they must be parallel. Since both lines πΆπ· and πΈπΉ are parallel to the same line, π΄π΅, we must have that the two lines are parallel. Hence, the answer is that line πΆπ· is parallel to line πΈπΉ, 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 πΏ one, πΏ two, and πΏ 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 πππΏ is equal to the measure of angle πππ, as they are vertically opposite angles. We now see that triangle πππ and triangle πππΏ are congruent by the ASA criterion. In particular, this means that πΏπ is equal to ππ. 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. Letβs now see some examples of using this property to find the length of a transversal between parallel lines.

If lines π΄π΅, πΆπ·, and πΈπΉ are all parallel and πΆπΈ equals two centimeters, find π΄πΈ.

We can see in the diagram that we are given three parallel lines and two transversals. We can also see that π΅π· is equal to π·πΉ. 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, π΄πΆ is equal to πΆπΈ, which is equal to two centimeters. We note that π΄πΈ is equal to π΄πΆ plus πΆπΈ. So π΄πΈ 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 π΄π΅πΆ and lines π΄π and πΈπ·, which are parallel to line πΆπ΅. Find the length of the line segment π΄π΅. Find the measure of angle π΄π΅πΆ.

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 π΄πΈ is equal to πΈπΆ, the segments of the other transversal must be equal in length. So π΄π· is equal to π·π΅, which equals five millimeters. Since π΄π΅ is equal to π΄π· plus π·π΅, then π΄π΅ equals five millimeters plus five millimeters, which equals 10 millimeters. π΄π΅ is equal to 10 millimeters.

It appears in the diagram that triangle π΄π΅πΆ 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 πΏ, then it is perpendicular to any line parallel to πΏ. Since line πΈπ· is perpendicular to line π΄πΆ and line πΈπ· is parallel to line π΅πΆ, we must have that lines π΅πΆ and π΄πΆ are perpendicular. This means the angle at πΆ has a measure of 90 degrees, so π΄π΅πΆ 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 π΄π΅πΆ. Rearranging the equation, we have the measure of angle π΄π΅πΆ 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.

Letβs finish by recapping some of the most important points from this video. If a line is perpendicular to line π΄π΅, then it is perpendicular to any line parallel to line π΄π΅. 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.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions