Angle Between Lines

Introduction 

In geometry, a line is a set point extending in the opposite direction with no ends on both sides. It may be of any thickness, and it is one-dimensional. 

There are various types of lines: horizontal lines, tangential lines, etc. Lines are infinite and do not have a limit; they may, however, be divided into segments by points. 

When one line intersects with another, they form an angle of intersection. 

An angle is a figure formed by two lines or rays that share a common endpoint. Derived from the Latin word “angulus,” angle means the corner of two intersecting lines in maths. 

The intersecting point of two lines makes an angle, where the two lines or rays are the angle’s sides, and the common endpoint is known as the vertex. Angle is denoted with the symbol ∠

The angle between two intersecting lines allows us to understand their relationship. It measures the inclination or slope of the two lines in relation to each other.

The angle of intersection between two lines can be calculated in several ways: it can be calculated through the slope of the two lines, by deriving the equations of the two lines, or by using the tangent function.

Formula to find the angle between two lines

Suppose there are two lines in a plane, l₁ and l₂, with the slopes denoted by m₁ and m₂, respectively. If the angle formed by these lines (inclinations) are θ₁ and θ₂, then by rule

m₁ = tan(θ₁) and

m₂ = tan(θ₂).

While considering a plane, when two lines intersect each other, they form a pair of vertically opposite angles and also angles in a linear pair. The sum of the adjacent angles (neighbouring angles) is complementary (180°).

Suppose that, α and β be the adjacent angles in between l₁ and l₂

α + β = 180° (i)

α= θ₂ – θ₁, such that θ₁,θ₂ ≠ 90°

This implies,

tanα = tan(θ₂ – θ₁)

tanα = (tanθ₂ – tan θ₁)/(1+ tan θ₁.tan θ₂)              [tan(A-B) = (tanA – tanB)/(1+ tanA.tanB)]

This can also be written as,

tanα =| m₂-m₁/1+m₁.m₂|

Also, β = 180°- α (From i)

tanβ = tan(180°- α) 

tanβ = -tanα [tan(180-A)= -tanA]

tanβ = m₂-m₁/1+m₁.m₂

This step results in two cases as follows:

Case 1: 

If the value of m₂-m₁/1+m₁.m₂  is positive, then tanα will also be positive, and tan β will be negative.

Thus, we can conclude that α is acute and β is obtuse.

Case 2:

If the value of m₂-m₁/1+m₁.m₂ is negative, then tanα will also be negative, and tanβ will be positive.

So, in this case, α is obtuse, and β is acute.

With reference to these cases, we can get the acute angle between two lines as

tanα = |m₂-m₁/1+m₁.m₂ | where 1+m₁.m₂ ≠ 0 (Considering α as acute angle)

If you need to calculate the obtuse angle between the two lines, you can use the formula β = 180°- α.

This formula allows us to easily derive the angle between lines regardless of whether they are acute or obtuse. This formula is useful in many different circumstances.

Using the formula to calculate the angle between lines

There are several conditions where you can use the angle between lines calculator to find the angle of intersection between two lines.

Example 1: Sometimes, the slopes of both intersecting lines are given in the problem. In that case, these steps may be followed to calculate the angle between the two lines. 

Question: If slopes of two lines are -2 and 6, find the acute angle between lines.

Given :Slopes, m₁= -2, m₂ = 6

To find : Acute angle (say θ )

Formula : tanθ =|m₂-m₁/1+m₁.m₂| 

Solution :

Using the formula,

tanθ = |m₁-m₂/1+m₁.m₂|

∴ tanθ = |{6-(-2)}/{1+ 6(-2)}|

∴ tanθ = |(6+2)/(1-12)|

∴ tanθ = |8/(-11)|

∴ tanθ = 8/11

∴ θ = tan^-1(8/11)

Hence, the acute angle between the line is tan^-1(8/11).

Example 2: Sometimes, however, the slopes of the lines are not provided. Instead, the endpoints of the lines and a point of intersection are given in the question. In that case, the following steps may be followed to calculate the angle between the two lines.

Question: Find the acute angle between lines with points (4,6) and (-3,7), meeting at point (2,2).

Given: 

Assume (4,6) coordinates of l1 and (-3,7) be the coordinates of  l2.

Therefore, (x₁,y₁) ≡ (4,6) and (x₂,y₂) ≡ (-3,7)

Point of intersection (x₃,y₃) ≡ (2,2)

To find : θ

Formula: tanθ = |(m₂-m₁)/1+m₁.m₂|

Slope of line (m)=y₂-y₁/x₂-x₁

Solution:

By formula,slope of l₁,

m₁ = (2-6)/(2-4)

∴ m₁ = -4/-2

∴m₁ = 2 (i)

Similarly,

Slope of l₂,

m₂ = (2-7)/{2-(-3)}

∴ m₂ = -5/(2+3)

∴ m₂ = -5/5

∴ m₂ = -1 (ii)

Now that we have calculated the slopes of both the lines, the acute angle between lines can be easily calculated by formula,

tanθ = |m₂-m₁/1+m₁.m₂|

∴tanθ = |(-1-2)/{1+ (-1).(2)}| 

∴tanθ = |-3/(1-2)|

∴tanθ = |-3/-1|

∴tanθ = 3

∴ θ = tan^-1(3)

Therefore, the acute angle between lines is tan^-1(3).

Conclusion

Many variations are possible in the questions that ask you to calculate the angle between two lines. Sometimes the slope value is given, while at other times, it must be calculated first. However, the formulae covered previously can be used to calculate the acute angle between two intersecting lines. 

This formula can also be used in other circumstances, such as when the acute angle between the intersecting lines and slope of one of the lines is given, and the slope of the second needs to be calculated.

When two lines intersect, their angle of intersection is critical. In geometry, it is important to know how to calculate these various values in different circumstances, where the information available is different. These angles are also useful in understanding the characteristics of geometrical objects like lines.