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Parallel lines are encountered in everyday life; for example, we notice parallel lines on railway tracks, zebra crossings, stairs, walls, window linings, lines on notebook pages, and so forth.
If two lines are drawn on the same plane, they are considered to be parallel. Let’s have a look at parallel lines and their attributes.
Parallel Lines Must Meet Certain Requirements
Parallel lines are defined as two straight lines on the same plane that do not intersect. They are also known as parallelograms.
The perpendicular distance between the two parallel lines is always the same, regardless of how many parallel lines there are.
Parallel lines are represented by the symbol ||, which is also known as ‘parallel to’.
Using a pair of vertical pipes between line names, such as m || n to denote parallel lines, it is possible to denote parallel lines between two or more lines.
Parallel lines are easy to memorise if you assume the “equal to (=)” symbol before each one. Some of the conditions for parallel lines are described in further detail farther down this page.
Parallel Lines Have Specific Characteristics
Parallel lines have a number of key characteristics, some of which are as follows:
The perpendicular distance between two parallel lines is always the same no matter where you look.
Two lines are parallel to each other if they are parallel to the same line, and all three lines are parallel to each other if they are parallel to each other
Whenever two lines are in the same plane and perpendicular to the same line, they are considered to be parallel to one another.
There is only one and only one parallel line that may be drawn parallel to a given line through a given point that does not lie on the provided line in question.
In this context, intersecting lines refer to lines that are not parallel to one another and cross each other at some point in their path. A transversal line is a line that intersects two or more lines at different locations during its length.
Transversal
A transversal of the supplied lines is a line that intersects two or more lines at various positions on the same side of the line. The transversal is indicated by the red line in the illustration below.
When the transversal intersects two lines in two points, it creates four angles at each point where the lines connect. There are a total of eight angles that have been constructed.
Because they are outside the lines, the angles ∠1, ∠2, ∠7, and ∠8 are referred to as the outside angles.
Because they are contained within the lines, the internal angles are numbered ∠3, ∠4, ∠5, and ∠6.
When a transversal connects two lines, it creates a pair of angles, which are as follows when they intersect:
A pair of angles on the same side of a transversal are known as corresponding angles if they both sit above or below the two lines on the transversal, respectively. The following are examples of corresponding angle pairs:
∠1 and ∠ 6 are the first and sixth positions, respectively.
∠4 and ∠7 are the most important numbers.
∠2 and ∠5 are the digits of the second and fifth positions, respectively.
the numbers ∠3 and∠ 8
Internal Angles on the Same Side of the Transversal: Two interior angles on the same side of the transversal are referred to as pairs of successive interior angles. The following are examples of interior alternate angles:
∠4 and ∠5 are the most important numbers.
∠3 and ∠6 are the most important numbers.
Exterior angles that alternate between
∠1, ∠8 and ∠2 .,∠7
Interior angles on the same side are ∠3 and ∠5.
∠4 and ∠ 6 are the most important numbers.
Now, let’s look at the relationships between the angles of the pairs when the two lines are parallel to each other and when the two lines are not parallel to each other, respectively.
Axioms and Theorems for Parallel Lines
Consider the line a||b, where l represents the transversal.
The axioms and theorems for parallel lines are given in the next section.
Angle Axiom with Corresponding Angle
The intersection of two parallel lines by a transversal results in the equality of each pair of corresponding angles between the two parallel lines.
For example, ∠1=∠6, ∠4=∠8,∠2=∠5, and ∠3=∠7 are all ∠6.
In contrast, if the angles of a pair of related angles are identical, then the lines in question are parallel to one another.
It is possible to establish the following theorem if we suppose that one of the assumptions is always true.
We can establish the following theorems by taking into account the corresponding angle axiom.
The first theorem is as follows
When two parallel lines are intersected by a transversal, the pair of alternate interior angles formed by the intersecting lines is equal.
For example, ∠4=∠5 and ∠3=∠6 are both correct.
We have, in fact.
∠4=∠2 as well as
∠1+∠3=∠4……….. (Vertically Opposite Angles)
∠2+∠5=∠10, and
∠1+∠6=……….. (Corresponding Angles Axiom)
As a result, we have
∠4 equals ∠5 and ∠3 equals ∠6.
For example, if two parallel lines are intersected by a transversal, the pair of alternate interior angles formed by the intersection are equal.
In contrast, if the pair of alternate interior angles is equal, then the two lines supplied are parallel to each other and vice versa.
Theorem 2
It is shown in Theorem 2 that when two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are additional.
∠3 +∠ 5 Equals 180 , and ∠4 +∠ 6 = 180, and ∠3 + ∠5 = 180
We have proof in the form of
∠4 +∠ 5 =∠ 6 and
∠3 + ∠6 =……….. (Alternate interior angles)
∠3+∠4=180 degrees, and
the sum of 5 and 6 equals 180° (Linear pair axiom)
∠3 + ∠5 equals 180°, and ∠4 + ∠6 equals 180°
The supplied lines are parallel to each other if the pair of co-interior angles are supplementary to each other.
These describe the parallel lines and their characteristics.
Conclusion
Lines that are parallel to each other can be seen in real life as well if one has the patience and the ability to notice them. Take, for example, railroads. The railway tracks go in a straight line. The train’s wheels are supposed to go along the two lines or tracks. The distinction between mathematicians’ parallel lines and those who actually build railway tracks is that mathematicians may conceive parallel lines on flat surfaces and paper, whereas trains run across a variety of terrain, including hills, slopes, mountains, and bridges.
