A Short Note On The Angle Between The Straight Line

The angle between two lines

The angle between the two lines helps to know the relationship between the two lines. This is a measure of the slope between the two lines. When two lines intersect, there are two angles between the lines: acute and obtuse. Here we consider the acute angle between the lines with respect to the angle between the two lines.

The angle between two lines can be calculated by knowing the slopes of the two lines or by knowing the equations of the two lines. The angle between two lines usually gives an acute angle between two lines.

The angle between the two lines can be calculated from the slope of the two lines using the trigonometric tangent function. Consider two straight lines with slopes of m1 and m2, respectively. The acute angle θ between straight lines can be calculated using the equation for the tangent function. The acute angle between the two lines is given by:

Tan Θ = m1 – m2 / 1 + m1m2

Formulas for the angle between two lines

for one line, which is ax + by + c = 0, and the other, which is the x-axis, the angle between the two lines is Θ = tan-1 (-a/b)

for one line, which is y = mx + c, and the other, which is the x-axis, the angle between the two lines is  Θ = tan-1 m

if the lines are parallel to each other and have equal slopes, then the angle between them is 0o

The angle between two perpendiculars with the same gradient product equal to -1 (m1m2 = -1) is 90o

The angle between a pair of straight lines

Given two straight lines y = m1x + c1 and y = m2x + c2, the angle between these two straight lines is given by tan θ = | (m1 – m2) / (1 + m1m2) |. 

 If m1, m2, and m3 are the slopes of the three lines L1 = 0, L2 = 0, and L3 = 0 (m1> m2> m3), the internal angle of the triangle ABC formed by these lines. It is given by the following formula

Tan A = (m1– m2) / (1 + m1m2)

Tan B = (m2– m3) / (1 + m2m3)

Tan C = (m3– m1) / (1 + m3m1)

Note: If one of the lines is parallel to the y-axis, the angle between the two lines is given by tan θ = ± 1 / m. Where “m” is the slope of another straight line. if a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 then we apply tan θ = | (a1 b2  – b1 a2 ) / (a1a2 + b1 b2 ) |. In general, the angle between these two lines is assumed to be an acute angle, so the value of tan θ is assumed to be positive.

Angle between pair of lines = ax2 + 2hxy + by2 = 0

By comparing the coefficients of x2, y2 and xy 

B(y – m1x) (y – m2x) = ax2 + 2hxy + by2

m1 + m2 = a/b

tan θ = | (m1 – m2) / (1 + m1m2) |

= tan θ = | √(m1 – m2)2 – 4 m1m2 / (1 + m1m2) |

= |2 √(h2 -ab) / (a + b) |

Conclusion

Two straight lines in a plane are parallel, coincident, or intersecting. When two lines intersect, they usually form two angles at the intersection. One is an acute angle and the other is an obtuse angle or higher. These two angles complement each other (equal to a total of 180 °). By definition, “angle between two lines” means an acute angle between two lines. The condition creates a linear relationship between any two constants, so any constant determines the other. Therefore, a row that meets the condition contains one arbitrary constant. Such a line system is called a parameter line family, and any unknown constant is called a parameter.