Angle Between Two Lines

An angle is generated when two rays or lines meet at a similar point, and each angle has a different measure. Acute angle, obtuse angle, right angle, reflex angle, and straight angle are all examples of angles in geometry.

The angle between two lines is beneficial for understanding the relationship between two lines. It’s the angle between the two lines that are being measured. There are two angles between two intersecting lines: the acute and obtuse angles. We consider the acute angle between the lines for the angle between two lines.

Two sets of angles are created when two straight lines intersect. The crossing establishes a pair of acute and another pair of obtuse angles. The slopes of crossed lines determine the absolute values of angles. It’s also worth noting that if one of the lines crosses the y-axis, the angle formed cannot be established because the slope of a line parallel to the y-axis is uncertain.

Method to Find the Angle Between Two Lines

Knowing the slope or the equation of the two lines can be used to compute the angle between two lines. The intersection of two lines forms the acute angle subscript between two lines.

The angle between two lines can be evaluated using the tangent function in trigonometry and the slope of the two lines.

Consider two lines, one with a slope of m₁ and the other with a slope of m₂. Then, the tangent function formula can calculate the acute angle between the lines. For example, the following formula calculates the acute angle between two lines:

Tanθ =( m₁-m₂)/1+m₁.m₂

If the equations of the two lines are known, we may calculate the angle between them.

For example, we will take the equation between the two lines as:

I₁= a₁x + b₁y + c₁ = 0 and I₂ = a₂x + b₂y + c₂ = 0

The tangent of the angle between the two lines can be used to calculate the angle between the two lines.

Tanθ =(a₂.b₁-a₁.b₂)/(b₁.b₂+a₁.a₂)

Formulas For Angle Between Two Lines

The following are some formulas that help in finding the angle between two lines:

● If one of the lines of the angle between two lines is ax + by + c = 0 and the other line is the x-axis, then θ= Tan^-1(a/b).

● If one of the lines of the angle between two lines is y= mx + c and the other lime is the x-axis, then θ= Tan^-1 m.

● The angle formed by two parallel lines with equal slopes crossing each other (m₁= m₂ ) is 0º.

● When two lines are perpendicular to each other and have the product of their slopes as -1 (m₁. m₂= -1), the angle between two lines would be 90°.

● The angle between two lines with m1 and m2 slopes is

θ =tan^-1( m₁-m₂)/1+m₁.m₂

●  Between a pair of straight lines ax²+ 2hxy + by² = 0, the angle would be  

Angle Between Two Lines In Three Dimensional Space

In three-dimensional space, the angle between two lines can be determined similarly to the angle between two lines in a coordinate plane.

If the two lines have these equations: r=a₁+λb₁ and r=a₂+λb₂, then the following formula is given for the angle between two lines:

In a 3D space, straight lines are usually represented in two ways: cartesian form or vector form. As a result, the angles formed by any two straight lines in the 3D space are specified in terms of both their forms.

Conclusion

Straight lines play an essential part in two-dimensional geometry in mathematics. A straight line is nothing more than the intersection of an unlimited number of points in two-dimensional space that stretch infinitely in either direction. As a result, a straight line (also known as a ‘line’) has no height and only length. In most circumstances, when we talk about the angle between two lines, we’re talking about the angle between two intersecting lines. The angle formed by two perpendicular lines is 90° (by definition), whereas the angle formed by two parallel lines is 0°.