Angle Between Intersecting Lines

The angle between the lines forms only when two or more lines intersect each other. The angle formed by two lines is beneficial to understanding the relationship between the two lines. It’s the angle between the two lines that are being measured. There are two angles between two intersecting lines that are, the acute angle and the obtuse angle, the acute angle is less than 90 degrees whereas the obtuse angle is greater than 90 degrees. For the angle between two lines, we consider the acute angle between the lines. By knowing the angle between two lines we can know whether they are perpendicular, parallel, or inclined with some other angle.

PROPERTIES OF INTERSECTING LINES: –

The properties of intersecting lines are in the following,

• Intersecting lines must meet at a single point i.e., they cannot intersect at several points
• Intersecting lines can cross each other at any angle between 0 and 180 degrees i.e. when two lines cross each other, the angle formed between them is from 0 to 180 degree

TO CALCULATE THE ANGLE BETWEEN THE GIVEN TWO LINES: –

 Knowing the slope of two or more lines or the equation of the two lines can be used to calculate the angle between them. The acute angle formed between two given lines is the angle formed by the intersection of those two lines. It is important to note that if one of the lines cut or intersect with the y-axis, the angle generated by the intersection cannot be determined because the slope of a line parallel to the y-axis is ambiguous.

The angle formed between two given lines can be calculated using the trigonometric tangent function and the slope of that two given lines. Consider two lines, one with a slope of m1 and the other with a slope of m2. The tangent formula is used to evaluate the acute angle between the lines. If the angle between the two given lines is θ. Then the following formula calculates the acute angle between the two lines. 

Tanθ = (m1.m2)/(1-m1.m2)

Where m1 is the slope of one line and m2 is the slope of another line and θ is the angle between the two lines.

Further, we can calculate the angle between two lines by knowing the equation of two lines. If the equations of the two lines are known, we can also calculate the angle between them. Let the two line’s equations be L1=A1X+B1Y+C1=0 and L2= A2X+B2Y+C2=0. Then the tangent of the angle between the two lines can be used to calculate the angle between the two lines.

Tanθ=(A 2B 1 -A1 B2 )/(A1A2+B1B 2)

In three-dimensional space, the angle between two lines can be determined similarly to the angle between two lines in a coordinate plane. For two lines with equations r= A1+λB1 and r=A2+λB2, the angle between two lines with equations is calculated using the formula below.

Cosθ=(B1.B2)/(|B1||B2|)

The formulas below can be used to quickly determine the angle between two lines.

• The angle formed by two lines, one of which is ax + by + c = 0 and the other is the x-axis, is equal to Tan ^-1 (-a/b).
• If y = mx + c is the one line and the x-axis is another line, then the angle between the given lines is θ = Tan^ -1 (m)
• The angle formed by two lines that are parallel and have equal slopes (m1 = m2) is 0°.
• The angle formed by two perpendicular lines with the product of their slopes equal to -1 (m 1. m 2 = -1) is the 90° angle.
• The angle formed by two lines with slopes of m 1 and m 2 is equal to Tanθ = (m1−m2)/ (1+m1.m2)
• The angle formed by two lines with equations L1=A1X+B1Y+C1=0 and L2= A2X+B2Y+C2=0 is  Tanθ=(A 2B 1 -A1 B2 )/(A1A2+B1B 2).

In daily life we use various applications of intersecting lines like the angle formed by two lines can be used to calculate the angle formed by two sides of a closed polygon. Intersecting lines are formed by the letters of the alphabet ‘X.’ It’s an excellent illustration of intersecting lines. (+) is the symbol for addition that is also the example of intersecting lines. Clock hands are the hands of a clock.

Point to Remember

• The Intersecting lines are formed when two or more lines cross each other or meet at a common point. The common point of intersection is the point at which they cross each other. By knowing the angle between two lines we can know whether they are perpendicular, parallel, or inclined with some other angle.