What are Parallel and Intersecting Lines?

The geometric approach and its pronounced is evident in the glimpses of our ancestral monuments and constructions. It gives evidence that geometry has existed since the time of human civilization ever started. The human race introduced the study of shapes and structure which came to be known as geometry.

Geometry is a subject of mathematics that deals with shapes, structures, and their properties.

Geometry gets its original name from the Greek words ‘ge’ and ‘materia’ which means earth and measurement respectively. By the time, the concept of geometry has spread way further as in today’s developing world geometry is not only limited to the study of infrastructure or structure of earth but it also applies in the field of technology, artificial intelligence, designing, graphics, and many more.

Lines

Lines are one-dimensional geometrical structures with uncountable numbers of points lying on them. A line has only one dimension that is length, this means a line does not comprise breadth or height. A line can extend infinitely into opposite directions as it does not have an endpoint. Hence, they are non-terminating. Those lines which consist of two endpoints are known as line segments and they are terminating in nature. Lines are categorized into different types based on the direction followed by them.

• Perpendicular lines
• Parallel lines
• Intersecting lines
• Curved lines, etc

Parallel Lines

The set of two or more than two lines that lies on the same plane at equal distance from each other and never intersect are known as parallel lines. Some real-life examples of parallel lines are railway tracks, lines in a notebook, zebra crossing, etc.

Properties of parallel lines

• Parallel lines are always placed at the same distance from each other.
• Parallel lines never intersect even though they are extended in either direction.
• In parallel lines, corresponding angles so formed are equal.
• In parallel lines, the alternate interior angles are equal.
• In parallel lines, alternate exterior angles are equal.

Formula

Suppose we have two equations of lines that are y = m₁x + c₁ and y = m₂x + c₂. Here, m₁ and m₂ are the slopes of the given line equation. So,

m₁ = m₂

It means both the lines are parallel to each other.

Intersecting lines

When two or more than two lines intersect or meet at a common point the lies are known to be intersecting lines. The particular point at which these lines meet each other is known as the point of intersection. Generally, all the lines moving in a particular direction intersects the other one at any point except for the parallel lines. And, these lines which do not intersect or meet at any point are known as non-intersecting lines.

Properties of intersecting lines

• Intersecting lines have a single point of intersection, which means they cannot meet at more than one point.
• The point of intersection lies at an angle that is greater than  0° and less than 180°.

Formula

Suppose we have two lines P and Q now we find the point of intersection(i.e., O). Now the equations of these two lines are:

P = a₁x + b₁y + c₁ = 0

Q = a₂x + b₂y + c₂ = 0

Let us assume the point of intersection is (x₁, y₁)

So, we get

a₁x₁ + b₁y₁ + c₁ = 0

a₂x₁ + b₂y₁ + c₂ = 0

Now using the Cramer’s rule we get

x₁/b₁c₂ – b₂c₁ = -y₁/a₁c₂ – a₂c₁ = 1/a₁b₂ – a₂b₁

So using this we get the point of intersection(x₁, y₁)

(x₁, y₁)= (b₁c₂ – b₂c₁/a₁b₂ – a₂b₁, c₁a₂ – c₂a₁/a₁b₂ – a₂b₁)

Sample Questions

Question 1: What are perpendicular lines?

Solution:

The pair of lines that meet each other at 90° making a right angle is known as perpendicular lines.

Question 2: What are non-intersecting lines?

Solution:

When two or more lines do not intersect each other at any point, these lines are known as non-intersecting lines.

Question 3: How can you know that the given pair of lines are parallel?

Solution:

We can determine whether the given pair of lines are parallel or not by their linear equations. As slopes of the, both lines are also equal in parallel lines along with the distance between them.

Question 4: Give some real-life examples of non-intersecting lines?

Solution:

Some real-life example of non-intersecting lines is opposite sides of a ruler, zebra crossing, railway tracks, etc.