Intersecting Lines

Last Updated : 26 Feb, 2024

Intersecting Lines are those lines which interact with each other at one point forming an intersection point. Also, at the point of intersection of two lines, 4 angles are formed. These angles form pairs of equal angles i.e. Vertical Opposite Angles. In this article, we will discuss Intersecting lines in detail.

Intersecting-Lines

What are Intersecting Lines?

Intersecting lines are lines that cross each other at a point. In geometry, when two lines intersect, they form angles at the point of intersection. The point where the lines meet is called the point of intersection.

Definition of Intersecting Lines

Intersecting lines are lines that meet or cross at a common point, known as the point of intersection, forming angles around this shared point.

Examples of Intersecting Lines

In everyday life, intersecting lines can be easily seen. For instance, the spokes of a bicycle wheel intersect at the hub and the lines of latitude and longitude on a map intersect to pinpoint locations. Some other examples of intersecting lines are:

Shape of a plus sign (+)
Corners of a rectangular windowpane
Crisscrossing roads on a city map
X marks on a treasure map
The structure of a tic-tac-toe game board
Railway tracks crossing each other at junctions
Gridlines on graph paper where they intersect.

Properties of Intersecting Lines

Some common properties of Intersecting lines are:

Intersecting Lines share a common point called the point of intersection.
They create four angles at the point of intersection.
- Two pair of opposite angles or two pair of adjacent angles
Intersecting lines can meet at any angle, from 0° to 180°, and they can only meet at one common point.
No two straight lines can meet at more than one point.
When two lines intersect each other, they form a pair of vertical angles that share a common vertex or the point of intersection and are faced opposite to each other. The vertical angles are always equal to each other.

Mathematical Representation of Intersecting Lines

For a₁x + b₁y = c₁ and a₂x + b₂x = c₂, then graph of both lines will intersect at one point i.e., point of intersection iff

a₁/a₂ ≠ b₁/b₂

Note: If a₁/a₂ = b₁/b₂, then we can check c₁/c₂ to verify further that lines are parallel or coincident.

Types of Intersecting Lines

Intersecting lines can be classified into different types based on their orientation and relationship to each other.

Perpendicular Lines
Non-Perpendicular Lines

Perpendicular Lines

Perpendicular lines are lines that intersect at a 90-degree angle.

They can be found in various real-life scenarios, such as:

Intersecting Roads: At a four-way intersection, the roads often meet at right angles, forming perpendicular lines.
Corners of Buildings: The corners of many buildings are constructed at right angles, creating perpendicular lines.
Shelves and Cabinets: In interior design, shelves and cabinets are often installed perpendicular to the walls.
Grids on Graph Paper: The horizontal and vertical lines on graph paper are perpendicular to each other.

Non-Perpendicular Lines

Any two pair of lines which are intersecting each other but angle between them is not 90° are called non perpendicular lines.

Some example of non-perpendicular lines in real life are:

The flight paths of airplanes in the sky, which can intersect at various angles depending on their direction and destination.
The lines of a soccer field, which intersect at angles that are not 90 degrees.
The lines of a basketball court, which also intersect at angles that are not 90 degrees.
The camera movements in space movies, which can follow non-perpendicular paths to create interesting shots.

Other than these there can some other relationships between two lines i.e.

Skew Lines: Skew lines refer to pairs of non-parallel straight lines in three-dimensional space that do not lie within a common plane.
Coincident Lines: Those lines which lies on top of each other i.e., same slope and same intercepts.

Intersection of Three Lines

The intersection of three lines can result in different configurations ranging from a single point of intersection to the formation of closed shapes like triangles. There are three possibilities of intersection of three lines:

Intersection At One Point
Intersection at Two Points
Intersection at Three Points

Intersection-of-Three-Lines

Theorems Related to Intersecting Lines

Some theorems which are related to the intersecting lines are:

Vertical Angle Theorem
Alternate Interior Angles Theorem

Let’s discuss the statement of these theorems.

Vertical Angles Theorem

Vertical Angles Theorem states that when two lines intersect, the vertical (opposite) angles are always equal (congruent) to each other.

Note: When two lines intersect, they form four angles. Among these, there are two pairs of nonadjacent angles known as vertical angles.

Alternate Interior Angles Theorem

When a transversal intersects two parallel lines, it creates several angles. Among these, the alternate interior angles are the ones formed on the opposite sides of the transversal but inside the parallel lines.

According to alternate interior angle theorem,

If a transversal crosses a set of parallel lines, the alternate interior angles are congruent.

Non-Intersecting Lines

Non-intersecting lines are lines that do not cross or intersect each other. They can be found in various fields such as mathematics, physics, and computer science.

Some examples of non-intersecting lines are:

Railway Tracks
Electrical Wiring
Pipelines

Properties of Non-Intersecting Lines

Some of the most common properties of non-intersecting or parallel lines are:

Non-intersecting lines never meet each other at any point, no matter how far they are extended.
The distance between two non-intersecting lines is always the same.
If a transversal line intersects two non-intersecting lines, certain pairs of angles formed:
- Corresponding angles
- Alternate interior angles
- Alternate exterior angles

Note: Corresponding angles, Alternate interior angles and Alternate exterior angles are always equal.

Read More about Properties of Parallel Lines.

Parallel and Intersecting Lines

Parallel and intersecting lines are two distinct types of lines in geometry. Parallel lines never intersect with each other, while intersecting lines meet at a common point.

Intersecting-and-Parallel-Lines

Some other differences between parallel and intersecting lines include:

Aspect	Parallel Lines	Intersecting Lines
Definition	Two or more lines that are equidistant from each other and never intersect.	Lines that meet or intersect at a common point.
Examples	Railway tracks, notebook lines, zebra crossings.	Crossing roads, intersecting lines on graphs.
Properties	Always at the same distance from each other. Corresponding angles are equal. Alternate interior angles are equal. Alternate exterior angles are equal.	Have a single point of intersection. Angle at the point of intersection lies between 0° and 180°
Formula	If two line equations are y = mx + c₁ and y = mx + c₂, then as both lines have same slope. Thus, both are parallel.	For a₁x + b₁y = c₁ and a₂x + b₂x = c₂, If a₁/a₂ ≠ b₁/b₂, then both lines have one point of intersection.

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