In geometry, a line is formed when two planes intersect each other. Parallel lines are defined as lines that lie in the same plane but can never meet. These lines exist in a single plane but they run parallel to each other. They never meet, touch or intersect each other at any point in space.

In the same way, a single line and a plane, or two planes that never share a point are called parallel.

You can note that the parallel lines definition states that for two lines to be considered parallel, they need to be on the same plane. So, if two lines exist in a three dimensional space but do not share a single point, such cannot be termed as parallel. These lines are termed as skew lines.

Parallelism is a property of affined geometries. Since, Euclidean space is a special case of affine geometries, parallel lines are discussed in Euclid’s parallel postulate. Other spaces such as hyperbolic space have similar properties called as parallelism.

The word parallel comes from the Greek word parallēlos. Here, para means beside and allēlōn, which is derived from allos, means of one another.

If it’s used as a noun, parallel means the way in which things are similar to each other. For example, you can draw parallels between the US invasion of Iraq and the Vietnam War. Parallel is also used for the imaginary lines drawn on the globe that are parallel to the equator.

However, in mathematics, the word parallel is used for two lines, a line and a plane or two planes that never intersect each other.

The two parallel lines must be equidistant from each other. Therefore, the distance formula will be unique to them.

Suppose equation (1) and (2) represent two lines p and q, which are neither parallel to x-axis nor y-axis.

y=mx+b1………(1)

y=mx+b2………(2)

Now, suppose that another line s intersects both of these lines. Furthermore, let line n be perpendicular to the parallel lines. If lines p and q have a slope “m” then line s must have a slope “-1/m.” So, its equation can be written as:

y=-xm

Now, solving the following systems:

If sides of a geometrical figure are parallel, it gives them a unique characteristic. If a shape has two parallel sides, think trapezoid, then:

- The parallel sides always form the base of the shape
- The height of the shape remains the same, regardless of the length

If two sides AB and CD are parallel, you can represent that mathematically by writing that AB // CD. Another way is to draw matching arrow marks on the parallel sides of the figure. If there are more than one parallel lines, add another arrow to mark the second pair. Similarly, two parallel sides can be marked as P // Q.

These conditions must be satisfied by two lines for them to be termed as parallel to each other.

Now, suppose that there are two lines l and m in the same Euclidean space. These lines are parallel to each other if:

- Every point of line m is at the same distance from a point on line l. That is, these lines must be equidistant at all points when extended to infinity.
- When a third line intersects both line l and m, the corresponding angles are congruent.
- Line l and m lie in the same plane but do not meet each other or share a common point.

All of the above properties are equivalent. So, all of these can be used for parallel lines definition. However, the first two properties require measurements in order to prove that two lines possess such properties. Therefore, the third one is often taken as parallel lines definition because of its simplicity.

Another possible parallel lines definition is that if two lines l and m are parallel, they will have the same slope or gradient.

Let’s discuss each of the four parallel lines definitions we mentioned above and take a look at which one is the most accurate one.

- Two lines are parallel if both are perpendicular to a third line

This definition is technically correct. The drawbacks, however, are that it supposes a third line which might not be a part of or stated in the situation. We will need to construct the third line, which is not a good way to define something in mathematics. There must be some intrinsic characteristic to prove a statement. Moreover, this definition supposes that the reader is already aware of perpendicular lines, which might not be the case. Therefore, it must not contain information about right angles and make it more complex.

- Two distinct lines are parallel if they never meet each other

This definition is similar to the classic example of a parallel line; the railroad track. Because the track goes on and on without meeting. Lines also extend in either direction up to infinity, therefore, the definition suits parallel lines definition well.

- Two distinct lines are parallel when they have the same slope

Now, this definition has two major drawbacks. The first one is that this definition applies to lines that have a slope. So, using this we cannot tell whether the lines parallel y-axis that is parallel or not. Therefore, not all possible solutions are satisfied.

This definition can be corrected by saying that “every vertical is parallel to each other, however, a vertical line can never be parallel to a non-vertical line.”

In addition to this, as there is no need to mention the slopes of two lines, this information in the definition will be treated as auxiliary information. To prove that two lines are parallel based on this definition, we will have to calculate the slopes that require coordinates. This requires that we construct the lines on a separate plane, something that Euclid never had with him. Moreover, this requires prior knowledge of slopes and their calculations.

Example 1:

If the slope of a line AB is x/4 and that of line CD is (x-5)/6. Find the value of x if lines AB and CD are parallel.

Solution:

Since lines AB and CD are parallel, their slopes are equal. So,

x4=x-56

6x=4x-20

2x=-20

x=-2

Example 2:

If the equation of a line AB is y=3x+2, and that of CD is 2y-3x=3. Find out whether they are parallel or not.

Solution:

Slope of line AB:

y=3x+2

y=mx+b

so m=3

Slope of line CD:

2y-3x=3

2y=3x+3

y=3x2+32

so

Since the slope of the lines are not equal, therefore, these are not parallel.

Example 3:

If lines AB and CD are parallel, and the slope of the lines are 3/4 and 8/(x-6). Find the value of x.

Solution:

For parallel lines, the slope of the two lines must be equal, therefore,

34=8x-6

3x-18=48

3x=66

x=22

Example 4:

What is the line parallel to a line having a slope 2/3.

Solution:

Here m=2/3 for line AB. It will be the same for line CD. So, the equation will be:

y=mx+b

y=23x+b

Example 5:

What is the slope of the line perpendicular to line AB, if the equation of line AB is given as: 3y-2x=6. Hint: m1 x m2= -1, where m1 and m2 are slopes of perpendicular lines.

Solution:

Slope of the line AB:

So,

3y-2x=6

3y=2x+6

y=23x+2

y=mx+b

For the perpendicular line, its slope will be m=-3/2. Therefore, its equation will become:

y=-32x+b

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