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# Angle Between Two Lines – Formula, Examples

angle between two lines helps to know the relationship between the two lines. is the measure of the slope between the two lines. for two intersecting lines, there are two angles between the lines, the acute angle and the obtuse angle. here we consider the acute angle between the lines, for the angle between two lines.

angle between two lines is useful for finding the measure of the angle between two sides of a closed polygon. Let’s see the formulas and examples for the angle between two lines in a coordinate plane and three-dimensional space.

The angle between two lines can be calculated by knowing the slope of the two lines or by knowing the equation of the two lines. the angle between two lines usually gives the acute angle between the two lines.

The angle between two lines can be calculated from the slope of the two lines and using the trigonometric tangent function. let us consider two straight lines with slopes (m_1), and (m_2) respectively. the acute angle θ between the lines can be calculated using the tangent function formula. the acute angle between the two lines is given by the following formula.

tanθ = (dfrac{m_1 – m_2}{1 + m_1.m_2})

Also, we can find the angle between the two lines if the equations of the two lines are given. be the equations of the two lines (l_1 = a_1x + b_1y + c_1 = 0), and (l_2 = a_2x + b_2y + c_2 = 0). the angle between the two lines can be calculated by the tangent of the angle between the two lines.

tanθ =(dfrac{a_2b_1 – a_1b_2}{a_1a_2 + b_1b_2}) The following different formulas help to easily find the angle between two lines.

• the angle between two lines, one of which is ax + by + c = 0, and the other line is the x-axis, is θ = tan-1(-a/b) .
• the angle between two lines, one of which is y = mx + c and the other line is the x-axis, is θ = tan-1m.
• the angle between two parallel lines with equal slopes ((m_1 = m_2)) is 0º.
• the angle between two lines that are perpendicular to each other and that have the product of their slopes equal to -1 ((m_1.m_2 = -1)) is 90º.
• the angle between two lines that have slopes (m_1), and (m_2) respectively is θ = (tan^{-1}dfrac{m_1 – m_2}{1 + m_1.m_2}).
• the angle between two lines that have equations ( l_1 = a_1x + b_1y + c_1 = 0), and (l_2 = a_2x + b_2y + c_2 = 0) is θ =(tan^{-1}frac{a_2b_1 – a_1b_2}{a_1a_2 + b_1b_2} )
• the angle between two lines that have equations (l_1 = a_1x + b_1y + c_1 = 0), and (l_2 = a_2x + b_2y + c_2 = 0) is cos θ = (frac{ a_ 1.a_2 + b_1.b_2 }{square root {a_1^2 + b_1^2}. sqrt{a_2^2 + b_2^2 }})
• the angle between a pair of lines ax2 + 2hxy + by2 = 0 is θ = (tan^{-1}frac { 2 sqrt {(h^2 – ab)}}{(a + b)})
• In a triangle that has sides of lengths, a, b, c, the angle between two sides of a triangle equals thing = (frac{b^2 + c^2 – a^2}{2bc})

The angle between two lines in three-dimensional space can be calculated in a similar way to the angle between the two lines in a coordinate plane. For two lines with equations (r = a_1 + λb_1), and (r = a_2 + λb_2), the angle between the lines is given by the following formula.

cosθ = (dfrac{b_1.b_2}{|b_1|.|b_2|})

Also, for two lines that have direction relationships like ((a_1, b_1, c_1)) and ((a_2, b_2, c_2)), the angle between the lines can be calculated using the following formula .

cosθ = (dfrac{a_1.a_2 + b_1.b_2 + c_1.c_2}{sqrt {a_1^2 + b_1^2 + c_1^2}. sqrt{a_2^2 + b_2^2 + c_2^2}})

Also for two lines that have direction cosines like (l_1, m_1, n_1) and (l_2, m_2, n_2), the angle between the two lines can be calculated using the following formula.

cosθ = (|l_1.l_2 + m_1.m_2 + n_1.n_2|)

related topics

The following topics help to clearly understand the concept of the angle between two lines.

• coordinate geometry
• equation of a line
• parallel lines
• types of angles