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Parallel Lines

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All around us we see many different forms of line, like the edge of the table, corners of floors and ceilings, sides of doors and windows, and so on. But there are some forms of straight lines which go side-by-side in the same direction **without intersection**. Like both opposite sides of the door in the same face. This type of line is an example of **parallel lines**.

In this explanation, we will understand the concept of **parallel lines** and their different properties.

Parallel lines are the types of lines that consists of two or more lines in the same plane.

Two or more straight lines in the same plane which are equidistant (having the same distance between them at all points) and never intersect each other at any point are called ** parallel lines**.

Parallel lines remain at the same distance from each other, no matter how far they are extended. They can be constructed in any direction whether horizontal, vertical, or diagonal. Mathematically, they are represented with the symbol which is called “ is parallel to”.

Here in the above figure p and q are parallel lines and m and n are parallel lines. Hence it is said thatandBut line a and line b are not parallel to each other as when extending both the lines, they both will intersect each other at some point. So a is not parallel to b (that is).

As parallel lines do not intersect with each other, no angles can be formed between them. But when another line apart from the given parallel lines intersect both the parallel lines then some angles are formed between them.

When any line cuts both the parallel lines at some point in the same plane, then this line is known as ** transversal**.

Here in the above figure, we can see that line *l* cuts both the parallel lines *a* and *b*. So line *l* is the transversal line. As the transversal cuts parallel lines it can be seen that the transversal forms pairs of angles with both the lines. There are different types of angles created by transversals.

The angles which are formed on the same side of the transversal and on the matching corners of parallel lines are called ** corresponding angles**.

Corresponding angles can be easily identified in the form of an “**F**” shape. They can be formed anyway either upside down or back and front. Corresponding angles are always equal to each other in parallel lines.

Angles formed on the opposite side of transversal on parallel lines are known as ** alternate angles**.

Alternate angles can be found in the form of a “**Z**” shape. They can be both interior and exterior angles. Similarly, like corresponding angles, alternate angles can be formed in any direction. Pair of alternate angles are always equal to each other.

Here in the above figure, both the angles are alternate interior angles.

Angles formed on the same side of the transversal facing each other on parallel lines are called ** interior angles**.

Interior angles are formed in the shape of a “U”. They can be found on either side of the transversal containing both parallel lines. The sum of interior angles will always be

Angles that are outside the sides of parallel lines but on the same of the transversal are called ** exterior angles**.

Exterior angles are formed in the shape of a “U” but will be located on the outside region of it. And the sum of the pairs of exterior angles will always be

Angles forming on any one of the parallel lines and transversals which are opposite to each other are called ** vertically opposite angles**.

Vertically opposite angles are found in the form of two “V” touching each other. They only contained any one of the parallel lines for each pair. Vertically opposite angles are equal to each other.

So we can represent all the pairs of angles for all the types of angles as below.

- Corresponding angle pairs :
- Alternate angle pairs :
- Interior angle pairs :
- Exterior angle pairs :
- Vertically opposite angle pairs :

Parallel lines are one type of line. So we can represent parallel lines in the form of an equation of the line. We know that in coordinate geometry, the equation of the line can be written in the form of So we can also represent parallel lines in the form of the equation

Here *b* is the y-intercept, so it can be any value. It is important to remember that as we have two or more lines in parallel lines the value of *b* for every line should be different from each other. As if they are equal then the equations of all lines will be the same and then it can be considered as one single line.

And *m* is the gradient or slope of that line. Here contrary to *b* the value of *m* for all the parallel lines should be equal. As *m* represents the slope of the line, if *m* is different for all parallel lines, then they will intersect each other and would not be considered parallel anymore.

We will understand the concept of gradient and how it can be found soon in the following topic.

The gradient or slope of parallel lines is the steepness of that line in the graph. The gradient of parallel lines is calculated with respect to the positive x-axis of the graph and parallel lines are inclined with the positive x-axis.

We know from above that the equation for parallel lines is Now suppose that the equation for one line and the equation for the other line is Here are y-intercept and *m* is the gradient of parallel lines. Then for both the lines to become parallel, the slope of both the lines should be equal. That is This equality can be derived by considering the angle between both the lines.

If we are already given two points on each line of the graph then we can calculate and verify the slope using the formula:

whereare the x-axis and y-axis points for the single line.

Let us see some parallel line examples and understand how to find angles and the slope in parallel lines.

In the given figure *m* and *n* are parallel lines and *l* is the transversal cutting both the parallel lines. Then find the value of *x *ifis given.

**Solution: **

We are already given that lines m and n are parallel to each other and line *l *is transversal to *m* and *n*.

So from the figure, we can clearly see that and are interior angles as they form the shape "U".

As both the angles are interior angles we know that their sum is equal to

Find the value offrom the given figure ifis given. Also line *a, b,* and *c* are parallel lines cut by the transversal *t*.

**Solution:**

It is given that lines *a,b* and *c* are parallel to each other, and line *t* acts as a transversal to these three lines.

First, we find the value of. We can see in the figure thatand angles forms a "U" shape. So both the angles and are interior angles. So the sum of both these angles will be

Nowandare vertically opposite angles. So both the angles will be equal to each other.

Now from the figure we can see that and forms a "F" shape, so they are corresponding angles. And hence they are equal to each other.

Hence the value of both angles are

Check whether the given lines are parallel lines or not.

**Solution:**

a) Here we are given two equations of lines Now comparing them with the general equation of parallel lineswe get that Hereare the gradients of parallel lines andare the y-intercepts.

As we know that for lines to be parallel, the gradients should be equal. And we can clearly see in the above equations that Also note that the valuesare different. Hence both the lines are parallel lines.

b) Here the equations of lines are given as Comparing it with the general equation of parallel lines we get that

As here we get thatwe can instantly say that both the given lines are not parallel to each other.

- Two or more straight lines in the same plane which are equidistant (having the same distance between them at all points) and never intersect each other at any point are called parallel lines.
- The angles which are formed on the same side of the transversal and on the matching corners of parallel lines are called corresponding angles.
- Angles formed on the opposite side of transversal on parallel lines are known as alternate angles.
- Angles formed on the same side of the transversal facing each other on parallel lines are called interior angles.
- Angles that are outside the sides of parallel lines but on the same of the transversal are called exterior angles.
- Angles forming on any one of the parallel lines and transversals which are opposite to each other are called vertically opposite angles.
- The equation of the parallel line iswhere the slope m of both the lines should be equal.

- Alternate angles should be equal
- Corresponding angles should be equal
- Vertically opposite angles should be equal.

Examples of parallel are railway tracks, opposite edges of doors and windows.

_{1} = m_{2}.

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