Lines and its Properties: Definitions, Types, Rules, Solved Examples

In this article, we will discuss in detail about Lines and its Properties. Geometry is the branch of mathematics that is associated with various geometrical figures. Most of the figures are formed by using the sides known as line segments. Thus, the line is the basic tool in geometry, in which the part of the line is called the line segment.

The line is the one-dimensional figure, which has only length, not width. There are different types of lines are there like parallel, perpendicular lines etc. Various properties are associated with the lines, such as it doesn’t have endpoints and extends infinitely in both directions.

Define Line

The line is the one-dimensional figure, which has only length, not width. According to Euclid’s second postulate, the line is a breadthless figure. A line has infinite points on it. The line extends infinitely in both directions. 

The length of the line is not fixed, as it has infinite length. The part of the line that connects any two points on it is called the line segment. Generally, line segments that are the parts of the lines are constructed by using the ruler. Lines are generally denoted by the arc over the points. The line is drawn between points \(A\) and \(B,\) extending it both sides infinitely, and it is denoted by \(\overleftrightarrow {AB}.\)

There are various types of lines in geometry, such as intersecting and non-intersecting lines, parallel lines, perpendicular lines, etc. The lines have a wide range of applications in mathematics.

Line, Line Segment and Ray

The line is the one-dimensional figure, which has only length, not width. According to Euclid’s second postulate, the line is a breadthless figure. The part of the line that connects any two points on it is called the line segment. The ray is the portion of the line that starts from any point and extends infinitely in any direction.

The below table gives the similarities and the differences between the line, line segment and the ray.

Line	Line Segment	Ray
A line is a breadthless length	The line segment is a part of the line with a fixed-length	A ray is the part of the line, which starts from one point and extends in one direction.
The line has no endpoints with infinite length	The line segment has two endpoints with a fixed-length	A ray has only one endpoint
The line can be represented by placing the arrow marks on both sides. Example: \(\mathop {AB}\limits^ \leftrightarrow \)	A line segment can be represented by placing the bar over the letters. Example: \(\overline {CD} \)	Ray can be represented by placing the arrow marks on one side. Example: \(\overrightarrow {EF} \)
A line can be extended infinitely in both the directions	A line segment can not be extended further; it lies between two points.	A ray extended in one direction.

Types of Lines

There are various types of lines based on the property that holds. The different types of lines are intersecting and non-intersecting lines, parallel lines, perpendicular lines, etc.

1. Intersecting Lines

Intersecting lines are the lines that formed when they cross each other at one point. The point of intersection is the point where the lines cross each other.

2. Non-Intersecting Lines

The lines that do not touch or intersect each other are known as non-intersecting lines. The non-intersecting lines are called parallel lines, and the distance between them is constant.

3. Parallel Lines

The lines are said to be parallel if they do not intersects each other and having the same distance between them from any point of the lines. Thus, parallel lines are non-intersecting lines having a constant distance. 

The railway tracks are the best example of parallel lines.

4. Perpendicular Lines

The lines are said to be perpendicular if they intersect each other at an angle of right angle \(\left( {{\rm{9}}{{\rm{0}}^{\rm{o}}}} \right){\rm{.}}\) The sides of the square and the rectangle are perpendicular lines. Perpendicular lines are the intersecting lines with an angle of \({{\rm{9}}{{\rm{0}}^{\rm{o}}}}.\)

5. Transversal Line

The line that intersects two or more parallel lines is called the transversal. In the image given below, the dotted line intersecting the two lines is called a transversal.

Properties of Line

The line is the one-dimensional figure, which has only length, not width. According to Euclid’s second postulate, the line is a breadthless figure. The various properties of the lines are listed below:

1. A line has infinite length.
2. The line extends in both directions infinitely.
3. The lines do not have any endpoints.
4. The line is a one-dimensional geometrical figure.
5. The line has only length, and it does not have any thickness.
6. Intersecting lines will cross at only one point.
7. The distance between the parallel lines is always the same.
8. The angle between the perpendicular lines is a right angle.
9. The part of the line between the given points is the line segment.
10. A line has infinitely many solutions.
11. The length of the common perpendicular drawn between the two non-intersecting lines is always the same.
12. The lines used in the graphs are used to locate the points and have many more applications, the \(x-\)axis is the horizontal line, and the \(y-\)axis is the vertical line.

When a transversal line intersects two parallel lines, there are eight angles formed. All these angles have some properties, which are discussed below:

1. The pair of corresponding angles are equal.
   Example: \(∠1=∠5, ∠4=∠8\)
2. The alternate interior angles are equal.
   Example: \(∠4=∠5\)
3. The pair of alternate exterior angles are equal.
   Example: \(∠1=∠8\) 
4. The co-interior angles are supplementary angles.
   Example: \(\angle 4 + \angle 6 = {180^{\rm{o}}}\)
5. The co-exterior angles are supplementary angles.
   Example: \(\angle 2 + \angle 8 = {180^{\rm{o}}}\)
6. Vertically opposite angles are equal.
   Example: \(∠5=∠8\)

Equation of the Line

The equation of the line passing through the point \(\left( {{x_1},{y_1}} \right)\) and having the slope \(“m”\) is given by \(\left( {y – {y_1}} \right) = m\left( {x – {x_1}} \right).\)

The equation of the line passing through the points \(\left( {{x_1},{y_1}} \right)\) and \(\left( {{x_2},{y_2}} \right)\) is given by \(\left( {y – {y_1}} \right) = m\left( {x – {x_1}} \right),\) where slope \(m = \frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}.\)

Learn Everything About Lines Here

Solved Examples– Line and its Properties

Q.1. Observe the figure shown below. Identify the lines, intersecting lines, parallel lines from the given figure.

Ans: We know that line is a one-dimensional figure which has the only length. The line extends infinitely in all directions. The figure shown above is the lines that extend infinitely.
Lines:
Line \(A,\) line \(B,\) line \(C\) and line \(D.\)
Parallel lines:
The lines that are not intersecting and have the same distance between them are the parallel lines. The parallel lines shown In the above figure are line \(A\) and line \(B.\)
Intersecting Lines:
The lines, which touch or intersects each other at one point are called the intersecting lines. Intersecting lines cross each other at one point. In the above figure, line \(A\) and line \(D,\) line \(A\) and line \(C,\) line \(B\) and line \(C,\) line \(B\) and line \(D,\) line \(C\) and line \(D\) are the intersecting lines.

Q.2. The below figure shows the variety of lines. Observe the figure and identify the anyone pair for each of parallel lines, perpendicular lines, transversal lines and intersecting lines.

Ans: Parallel lines:
The lines that are not intersecting and having the same distance between them are called parallel lines. In the above figure, the lines \(AB\) and \(CD,\) lines \(CD\) and \(EF\) are parallel lines, as they do not intersect each other.
Perpendicular lines:
The perpendicular lines are the lines having the angle between them is equal to the right angle. In the above figure, the lines \(AB\) and \(PQ,\) line \(CD\) and \(PQ\) are the perpendicular lines.
Transversal lines:
The line that intersects the parallel lines are called the transversal line. In the above figure, the lines \(PQ, KL\) and \(MN\) are the transversal lines.
Intersecting lines:
The lines that touch or intersects each other are called the intersecting lines. In the above figure, the lines \(KL\) and \(MN\) are the intersecting lines, as they have to intersect at point \(O.\)

Q.3. Identify the pair of lines given below as intersecting or non-intersecting lines.

Ans: If we extend the lines further in the same direction, the lines intersect each other. So, the given pair of lines are intersecting lines.

Q.4. Identify the point of intersection of the lines shown in the figure.

Ans: The common point between two intersecting lines is called the point of intersection. The point of the intersections of the lines \(WX\) and \(YZ\) is \(M,\) for the lines \(UV\) and \(YZ\) is \(N,\) for the lines \(UV\) and \(WX\) is \(O.\)

Q.5. Find the equation of the line passing through the point \((2, 3)\) and having the slope of \(5\) units.
Ans: We know that the equation of the line passing through the point \(\left( {{x_1},{y_1}} \right)\) and having the slope \(“m”\) is given by \(\left( {y – {y_1}} \right) = m\left( {x – {x_1}} \right).\)
\( \Rightarrow (y – 3) = 5(x – 2)\)
\( \Rightarrow y – 3 = 5x – 10\)
\( \Rightarrow 5x – y = 7\)
Hence, the equation of the line is \(5x – y = 7.\)

Summary

In this article, we have discussed the definitions of line, line segment and the ray. We also discussed the difference between them. In this article, we have studied the types of lines and the various properties of the lines. We also referred to the solved examples to understand the concept of line and its properties easily.

Frequently Asked Questions

Q.1. What is the line in geometry?
Ans: The line is the one-dimensional figure, which has only length, not width.

Q.2. What are the intersecting lines?
Ans: Intersecting lines are the lines that intersect or cut each other at one point.

Q.3. Does a line have any endpoints?
Ans: No. The line does not have any endpoints.

Q.4. How is a line segment different from a line?
Ans: The line extends infinitely in both directions, and the line segment is of fixed length.

Q.5. How do you check the perpendicular lines?
Ans: The lines, which are intersecting at a right angle are called perpendicular lines.

Learn About Intersecting And Non-Intersecting Lines

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