Angle Between Two Lines

Whenever two straight lines intersect, they form two sets of angles. The intersection forms a pair of acute and another pair of obtuse angles. The absolute values of angles formed depend on the slopes of the intersecting lines.

It is also worth noting here that the angle formed by the intersection of two lines cannot be calculated if one of the lines is parallel to the y-axis as the slope of a line parallel to the y-axis is an indeterminate.

Angle Between Two Straight Lines Formula

If θ is the angle between two intersecting lines defined by y₁= m₁x₁+c₁ and y₂= m₂x₂+c_2, then, the angle θ is given by

tanθ=±(m₂-m₁) / (1+m₁m₂)

Angle Between Two Straight Lines Derivation

Consider the diagram below:

In the diagram above, the line L₁ and line L₂  intersect at a point.

Let the slope measurement can be taken as

tan a₁ = m₁ and tan a₂ = m₂

Also, from the figure, we can infer that θ = a₂-a₁

Now, tan θ = tan (a₂-a₁) = (tan a₂ – tan a₁ ) / (1- tan a₁tan a₂)

Substituting the values of tan a₁ and tan a₂ as m₁ and m₂ respectively, we have,

tanθ= (m₂ – m₁ ) / (1+m₁m₂)

It should be noted that the value of tan θ in this equation will be positive if θ is acute and negative if θ is obtuse.

Angle Between Two Lines Coordinate Geometry

In analytic geometry, if the coordinates of three points A, B, and C are given, then the angle between the lines AB and BC can be calculated as follows:

For a line whose endpoints are (x₁, y₁) and (x₂, y₂), the slope of the line is given by the equation

m =

The angle between the two lines can be found by calculating the slope of each line and then using them in the formula to determine the angle between two lines when the slope of each line is known from the equation

tan θ=± (m₁ – m₂ ) / (1+ m₁m₂)

We shall explore solved numerical problems in the next section.

Problem With Solution

If P (-2, 1), Q (2, 3) and R (-2, -4) are three points, find the angle between the straight lines PQ and QR.

The slope of PQ is given by

m = ( y₂ – y₁ ) / (x₂ – x₁)

m =( 3 – 1 ) / (2 – (-2 ))

m= 2/4

Therefore, m₁=1/2

The slope of QR is given by

m= (−4−3) / (−2−2)

m= 7/4

Therefore, m₂ = 7/4

Substituting the values of m₂ and m₁ in the formula for the angle between two lines when we know the slopes of two sides, we have,

tan θ=± (m₂ – m₁ ) / (1+m₁m₂)

tan θ=± ((7/4) – (1/2) ) / (1+ (1/2)(7/4))

tan θ=± (2/3)

Therefore,  θ = tan ^-1 (⅔)

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