In geometry, intersecting lines are two or more lines that cross or meet at a single point. This meeting point is called the point of intersection. Lines go on forever in both directions, so they don’t have starting or ending points. That’s why we show lines with arrows on both ends.

On the other hand, a line segment has a fixed length and has two endpoints.

Most lines in geometry will eventually meet or intersect, except for parallel lines, which never cross, and skew lines, which do not lie in the same plane and never meet.

When two lines intersect, they usually form an angle at the point where they meet. This angle can be measured and is important in geometry for solving many problems.

Intersecting lines are commonly seen in everyday objects like roads crossing or scissors opening and closing.

In this Maths article, we will look into the angle between two lines definition, formulas, steps and solved examples.

Angle Between Two Lines

When two lines cross or meet each other, they form angles at the point where they intersect. These angles help us understand the relationship between the two lines. Usually, two types of angles are formed: one is an acute angle (less than 90°), and the other is an obtuse angle (more than 90°). These two angles always add up to 180°.

The angle between two lines tells us how steeply they cross each other. To find this angle, we can use the slopes (or steepness) of the lines. There is a special formula in mathematics using the tangent function (tan) to calculate the angle when the slopes of both lines are known.

Understanding these angles is important in geometry, especially when working with graphs, designs, and real-life objects where lines cross each other.

\(\tan \theta = \left | \frac{m_{2}-m_{1}}{1+m_{1}m_{2}} \right |\)

Angle Between Two Lines Formulas

The many formulas listed below make it simple to determine the angle between two lines.

Steps to find Angle Between Two Lines

For example: To find the angle between the lines 2x-3y+7 = 0 and 7x+4y-9 = 0, use the formula.

Solution:

Step: 1

First, Comparing the equation with equation of straight line, y = mx + c,

Step: 2

Find the Slope of line 2x – 3y +7 = 0 is \(\left ( m_{1} \right ) = \frac{2}{3}\)

Step: 3

Find the Slope of line 7x + 4y + 9 = 0 is \(\left ( m_{2} \right ) = \frac{-7}{4}\)

Step: 4

Suppose \( \theta \) is the angle formed by two lines.

\( \tan \theta = \pm \frac{m_{1}-m_{2}}{1+m_{1}m_{2}}\)

\( \tan \theta = \pm \frac{\frac{2}{3}+\frac{7}{4}}{1- \frac{2}{3}.\frac{7}{4}}\)

\( \tan \theta = \pm \frac{8+21}{12-14}\)

\( \tan \theta = \pm \frac{29}{-2}\)

\(\tan \theta = \tan^{-1}\left ( \pm \frac{29}{2} \right )\)

Angle between Two Lines in Three Dimensional Space

Similar to the calculation of the angle between two lines in a coordinate plane, the angle between two lines in a three-dimensional space can also be determined.

Two lines containing equations \( r = a_{1} + \lambda b_{1}\) and \(r = a_{2} + \lambda b_{2}\)

The formula below provides the angle between the lines.

\(\cos \theta = \frac{b_{1}.b_{2}}{\left | b_{1}\right |.\left | b_{2} \right |}\)

The following formula can be used to calculate the angle between two lines whose direction ratios are \( \left ( a_{1},b_{1},c_{1} \right )\) and \(\left ( a_{2},b_{2},c_{2} \right )\)

\(\cos \theta = \frac{a_{1.}a_{2}+b_{1}.b_{2}+c_{1}.c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}.\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{3}}}\)

Angle between line and plane

A straight line has one dimension that is measured in terms of length, but a plane has two dimensions that are measured in terms of length and width. When a line comes into contact with a plane, the two lines create an angle, which is referred to as the angle between a line and a plane.

Given the direction vector of line L in space

\bar{s} = \left \{ l;m;n \right \}

as well as a plane equation

Ax + By +Cz + D = 0

Hence this formula can be used to determine the angle between this line and plane.

\( \sin \phi = \frac{\left | A.l+B.m+C.n \right |}{\sqrt{A^{2}+B^{2}+C^{2}}.\sqrt{l^{2}+m^{2}+n^{2}}}\).

Angle Between Two Lines – Coordinate Geometry 

In coordinate geometry, if you know the coordinates of three points A, B, and C, you can find the angle between the lines AB and BC.

To do this, first find the slope of each line:

For a line passing through two points (x1, y1) and (x2, y2), the slope (m) is:
m = (y2 - y1) / (x2 - x1)

Once you find the slopes of both lines (say m1 for line AB and m2 for line BC), you can find the angle between them using this formula:

tan(theta) = |(m1 - m2) / (1 + m1 * m2)|

This gives you tan(theta), and you can find the angle theta by using a calculator:

theta = tan inverse (value)

This method helps to calculate the angle where two lines meet, based on their slopes.

Solved example

Example 1: If there are three points P(–2, 1), Q(2, 3), and R(–2, 4), find the angle between the lines PQ and QR.

Solution:

We are given three points:
P(–2, 1), Q(2, 3), and R(–2, 4)

Step 1: Find the slope of line PQ

Use the slope formula:
m = (y₂ – y₁) / (x₂ – x₁)

Here, P(–2, 1) and Q(2, 3)

So,
m₁ = (3 – 1) / (2 – (–2)) = 2 / 4 = 1/2

Step 2: Find the slope of line QR

Points Q(2, 3) and R(–2, 4)

m₂ = (4 – 3) / (–2 – 2) = 1 / (–4) = –1/4

Step 3: Use the angle formula between two lines

tan(θ) = |(m₁ – m₂) / (1 + m₁·m₂)|

Substitute m₁ = 1/2 and m₂ = –1/4:

tan(θ) = |(1/2 – (–1/4)) / (1 + (1/2)(–1/4))|
= |(1/2 + 1/4) / (1 – 1/8)|
= |(3/4) / (7/8)|
= 3/4 ÷ 7/8 = (3/4) × (8/7) = 24/28 = 6/7

Step 4: Find the angle

θ = tan⁻¹(6/7) ≈ 40.60°

The angle between lines PQ and QR is approximately 40.60 degrees.

Example 2: Find the angle between two straight lines with equations:
2x + 3y - 5 = 0 and 3x - 4y + 1 = 0

Solution:
We are given two line equations:
Line 1: 2x + 3y - 5 = 0 → Here, a₁ = 2, b₁ = 3
Line 2: 3x - 4y + 1 = 0 → Here, a₂ = 3, b₂ = -4

To find the angle between the two lines, we use the formula:

tan(θ) = |(a₂b₁ - a₁b₂) / (a₁a₂ + b₁b₂)|

Substitute the values:

tan(θ) = |(3×3 - 2×(-4)) / (2×3 + 3×(-4))|
tan(θ) = |(9 + 8) / (6 - 12)|
tan(θ) = |17 / (-6)| = 17/6

Now find the angle:

θ = tan⁻¹(17/6) ≈ 70.34°

The angle between the two lines is approximately 70.34°.

Example 3: Find the angle between two straight lines with equations:
x + 2y - 7 = 0 and 2x - 3y + 4 = 0

Solution:
We are given two line equations:
Line 1: x + 2y - 7 = 0 → a₁ = 1, b₁ = 2
Line 2: 2x - 3y + 4 = 0 → a₂ = 2, b₂ = -3

Use the formula to find the angle between two lines:

tan(θ) = |(a₂b₁ - a₁b₂) / (a₁a₂ + b₁b₂)|

Substitute the values:

tan(θ) = |(2×2 - 1×(-3)) / (1×2 + 2×(-3))|
tan(θ) = |(4 + 3) / (2 - 6)|
tan(θ) = |7 / (-4)| = 7/4

Now calculate the angle:

θ = tan⁻¹(7/4) ≈ 60.26°

The angle between the two lines is approximately 60.26°.

If you want to score well in your math exam then you are at the right place. Here you will get weekly test preparation, live classes, and exam series. Download the Testbook App now to prepare a smart and high-ranking strategy for the exam.