Parallel lines or parallelism are a basic concept within geometry. In this article we are going to explain what parallel lines or parallelism are and their meaning. We will also teach you different ways to know if two lines are parallel or not. If you are ready to learn more about parallel lines and parallelism, be sure to read the following post. Go for it!
What are parallel lines or parallelism? What is its meaning
Parallel lines
The term parallel line is composed of two words. Straight and parallel. Both with a specific definition.
Definition of line
When speaking of lines in geometry, reference is made to a succession of points that have no end and extend in the same direction. The characteristic that precisely defines lines is their inexistence of beginning or end and their infinite form. These elements are precisely what distinguishes a line from:
- A half line, which unlike the line does have a beginning but does not have a defined end.
- A segment. The segment, however, does have a beginning and an end marked at a specific point.
Definition of parallel
When we refer to something parallel, we are talking about an object, whatever it may be, that is kept at an equidistant distance from a different object. These elements, even if they are infinite, will never intersect with each other. In case of doing so, they would no longer be parallel objects. For two objects to be parallel, they do not have to be identical. Simply that they are equidistant from each other and that they never touch each other. For example, two streets can be parallel, a glass and a jug can also be placed parallel, and so on.
We will consider that two lines are parallel as long as:
- Their direction vectors are parallel.
- If their direction vectors are equal.
- If both have the same slope.
- When the coefficients of Y and X are proportional to each other.
- When between the two they form angles of 0º.
Now that we are clear about what a line is and what parallel means, we can more easily understand what a parallel line is.
Parallel lines are those that, although infinite, never cross each other within the same plane. In addition, they are always separated at an equidistant distance and have the same slope. Have you ever noticed the train tracks? Have you noticed that the tracks seem to go on forever without ever touching and always with the same slope? This is so because they are composed of parallel lines.
Properties of Parallel Lines
Parallel lines can also be:
- Symmetrical parallel lines. When line A is parallel to line B, line B is also parallel to line A. That is, they both have symmetrical properties.
- Reflexive parallel lines. Parallel lines are always parallel to themselves. Hence they all have the reflexive property.
- Transitive parallel lines. A very interesting curiosity. If line A is parallel to line B and line B is parallel to C, then both A and C will be parallel to each other. And so it will happen successively. This is what it means for them to be transitive.
- Within the transitive property that we have just explained, another property is also added. If we already know that if two lines are parallel to each other, a third one that is parallel to one of them will also be parallel to the first one, we can add another property to it. All these lines will go in the same direction. This is what we mean by the corollary of the transitive property.
Parallelism
When we talk about parallelism within the field of geometry we are referring to the relationship that exists between the linear varieties that exist. Parallelism can occur in planes, between lines, between hyperplanes, and so on. Let’s see some examples.
- When we talk about parallelism between planes, a simple example to understand it is to think of sheets of paper that we have placed one on top of the other. These would have a parallel relationship with each other.
Tricks to check if two lines are parallel
We already have a more or less clear explanation of parallel lines and parallelism. We know that these are lines that never cross each other in a plane and that maintain the same slope. Therefore, a trick to check if two lines are parallel is to find out if they have the same slope. To do this we must compare the slopes of these lines.
What do we mean when we talk about slope? It is the inclination that an element has, in this case the lines, in reference to the horizontal axis. What marks the slope is the level of inclination of the line.
When we want to represent parallel lines, we do it using two vertical lines in a row. In this way, observe: (II)
Thus, if we do not find the following reference on a piece of paper: CD II AB. It will mean that CD is parallel to AB.
Well, now that we know how parallel lines are represented, we are going to check if you have really understood how these lines are represented. To do this, we are going to propose a series of very fun visual exercises where you will be able to find out if you have really understood what these lines are like. Do you want to start the challenge? In that case, keep reading!
Exercises on parallel lines or parallelism
To make it even easier for you, we have divided the exercise section into two parts. On the one hand the statement of the exercises and on the other the solutions. In the statement you will find the problem and the images that you have to observe to know what type of line we are talking about. In the solutions section you can check if your answers were correct. Let’s get started!
Statement of the exercises on parallel lines or parallelism
- Exercise 1. Rocío and trains Rocío loves trains. Since she turned eight years old, there is nothing she likes more than going with her parents to the station to see how they pass. Even riding them is really fun. Her parents love to see how her daughter enjoys watching the trains. And for this reason they have wanted her experience to be even more fun and educational. For this they have proposed a game. Look at the train tracks and find out what kind of lines they create. Looking at these photos, could you tell us what type of lines we are facing?
- Exercise 2. The curious case of the scissors. Marcos loves doing research around the house. He dreams that when he grows up he will become a great detective. He even has a detective cap and a magnifying glass that he plays with. And today is a lucky day because he can solve one of his cases. His client is dad and the case is The Curious Case of the Scissors. Daddy is going to start sewing the hem of mommy’s pants and he has thread, a meter stick and some nice scissors handy. While he does the sewing he has decided to ask Marcos a question. “The blades of the scissors are composed of two straight lines. If you look at the lines they have a relationship with each other. Could we say that they are parallel?” Do you dare to find out the answer together with Marcos?
- Exercise 3. The clueless pedestrian. Julio is visiting Madrid. He has decided to travel to the capital of Spain to visit his uncles. He likes this city very much because it is big and has beautiful monuments. And he loves the endless number of crosswalks that he finds around him. At most traffic lights and at many crossroads there is a zebra crossing. Have you ever noticed the drawing made by the zebra crossing? Have you seen that there are many straight lines one after the other, with the same slope and the same distance between them? Can you tell what kind of lines they are?
Solutions
- Solution to exercise number 1. Rocío and the trains. Are you as excited about trains as Rocío? Do you like to travel within your city, or from one city to another, in them? We are sure yes. But let’s not deviate from the topic. In the image of exercise number 1 we have shown you the rails of a train track. These straight-like rails are parallel. You know why? Well, very simple, because they have the same slope, they never touch each other and they move in the same direction. Therefore, they fulfill all the rules to be considered parallel lines.
- Solution to exercise number 2. The curious case of the scissors. We are sure that Marcos will become a great detective when he is older. I’m sure he’s just as famous as the detectives that came from the imagination of many writers. Like Hercule Poirot or Sherlock Holmes for example. But let us return to the curious case of the scissors. Could you tell us what type of straight lines are formed by scissors? In this case it is about oblique secant lines. Why? Well, because they intersect at one point and do not create 90º angles or right angles. If they had created right angles to each other, they would have been perpendicular secant lines.
- Solution to exercise number 3. The clueless pedestrian. We understand that Julio is easily confused when he visits Madrid. This great city is full of beautiful zebra crossings through which citizens can cross. Thanks to these zebra crossings you can go from one sidewalk to another and go through different streets. However, do you know what kind of lines make up the lines of the zebra crossing? Yes! They are parallel lines. Why? Very simple because they do not touch or cross at any time. They have the same slope and are moving in the same direction.