Parallel lines are lines that never intersect. These lines are characterized by being equidistant at each corresponding point. We can determine whether two or more lines are parallel by ensuring that their slopes are the same.
Here, we will learn more details about parallel lines and solve some practice examples.
What are parallel lines?
Parallel lines are defined as lines that lie on the same plane and do not touch (intersect) one another. For example, in the following diagram, we can observe pairs of parallel lines. That means the line pairs will never touch each other no matter how long we extend them.
Lines do not necessarily have to be straight to be parallel. For the lines to be parallel, they must be equidistant along their entire length. Curved lines can also be parallel as long as they remain equidistant and do not run into each other.
Angles of parallel lines
When we have two parallel lines that are crossed by a third line, called the transversal, several angles are formed. The angles formed have unique characteristics and several of them are the same to each other. In total, 8 angles are formed and depending on the different pairs of equal angles, they have different names that characterize them.
In the diagram, we can see each of the 8 angles formed. Each angle has a different letter to denote them. In addition, in the following table, we can observe the different pairs of angles:
|Corresponding angles||a=e, b=f, c=g, d=h|
|Alternate interior angles||c=e, d=f|
|Alternate exterior angles||a=g, b=h|
|Vertical angles||a=c, b=d, e=g, f=h|
Also, any pair of consecutive angles are supplementary, that is, they add up to 180°. For example, angles b and a add up to 180°, as do angles a and d or angles d and c.
Equation of parallel lines
The equation of a line, in general, can be written in its point-slope form: , where m is the slope and b is the y-intercept. The value of m determines the slope or steepness of the line. This value indicates the change in y over the change in x of the line.
For two lines to be parallel, the value of their slopes must be the same. Also, the parallel lines must have different values of b and not have any points in common.
In the following diagram, we can see two parallel lines with their respective equations. Specifically, notice that the value of the slopes is the same and that the value of the y-intercepts is different.
Parallel lines symbol
Parallel lines are lines that never touch each other no matter how much we extend them. We can denote parallel lines using the symbol ||. For example, if lines AB and CD are parallel, we can write AB||CD.
Additionally, we have other symbols that we can use. For example, the symbol ∦ means that the lines are not parallel. And the symbol ⋕ indicates that the lines are parallel and equal.
How to know if to lines are parallel?
There are two main methods that we can use to determine if two lines are parallel.
Using the slopes
In the case that we have the equations of the lines or that it is easy to determine the equations of the lines, we can use their slopes to find out if two lines are parallel. For the lines to be parallel, they must have the same slope.
Furthermore, the y-intercept must be different since, otherwise, we will obtain the same line.
Using the angles formed
We can use the angles formed by the parallel lines and a transverse line that crosses both lines. Two lines will be parallel if one of the following conditions is met:
- Any pair of corresponding angles are equal
- Any pair of alternate interior angles are equal
- Any pair of alternate exterior angles are equal
- Consecutive internal angles are supplementary
Solved problems of parallel lines
The following are some exercises for applying parallel lines.
Find the slope of a line parallel to the line .
Solution: We know that two lines are parallel if they have the same slope. Therefore, we have to determine the slope of the given line.
We write the line in its point-slope form. That means we have to solve for y:
The slope of the line is . Therefore, the slope of the parallel line must also be .
In the following diagram, the lines and are parallel, that is, we have ||. Furthermore, we also have ∠a=25°. What is the measure of angle b?
Solution: Since the lines and are parallel. Angles a and b are supplementary. Therefore, we have:
∠b = 180°-25° = 155°
Angle b measures 155°.
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