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Angle Between Two Lines
The angle between is the measure of the inclination between the two lines. For two intersecting lines, there are two types of angles between the lines, the acute angle and the obtuse angle. Here we consider the acute angle between the lines to be the angle between two lines.
The angle between two lines whose slopes are m_{1} and m_{2} is given by the formula tan^{1}(m_{1}  m_{2})/(1 + m_{1} m_{2}). Let us check all the formulas related to the angle between two lines in a coordinate plane and threedimensional space.
1.  How to Find Angle Between Two Lines? 
2.  Formulas For Angle Between Two Lines 
3.  Angle Between Two Lines in Three Dimensional Space 
4.  FAQs on Angle Between Two Lines 
How to Find Angle Between 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 generally gives the acute angle between the two lines.
The angle between two lines can be computed from the slope of the two lines, and by using the trigonometric tangent function. Let us consider two lines with slopes \(m_1\), and \(m_2\) respectively. The acute angle θ between the lines can be calculated using the formula of the tangent function. The acute angle between the two lines is given by the following formula.
tan θ = \(\left\dfrac{m_1  m_2}{1 + m_1m_2}\right\)
Further, we can find the angle between the two lines if the equations of the two lines are given. Let the equations of the two lines be \(l_1 = a_1x + b_1y + c_1 = 0\), and \(l_2 = a_2x + b_2y + c_2 = 0\). The angle between the two lines can be computed by the tangent of the angle between the two lines.
tan θ =\(\left\dfrac{a_2b_1  a_1b_2}{a_1a_2 + b_1b_2}\right\)
Formulas for Angle Between Two Lines
The following different formulas help in easily finding the angle between two lines.
 The angle between two lines, of which, one of the line is ax + by + c = 0, and the other line is the xaxis, is θ = tan^{1}(a/b).
 The angle between two lines, of which one of the line is y = mx + c and the other line is the xaxis, is θ = tan^{1}m.
 The angle between two lines that are parallel to each other and have equal slopes (\(m_1 = m_2\)) is 0º.
 The angle between two lines which are perpendicular to each other and have the product of their slopes equal to 1 (\(m_1m_2 = 1\)) is 90º.
 The angle between two lines having slopes \(m_1\), and \(m_2\) respectively is θ = \(tan^{1}\left\dfrac{m_1  m_2}{1 + m_1m_2}\right\).
 The angle between two lines having equations \(l_1 = a_1x + b_1y + c_1 = 0\), and \(l_2 = a_2x + b_2y + c_2 = 0\) is θ =\(tan^{1}\left\frac{a_2b_1  a_1b_2}{a_1a_2 + b_1b_2}\right\)
 The angle between two lines having equations \(l_1 = a_1x + b_1y + c_1 = 0\), and \(l_2 = a_2x + b_2y + c_2 = 0\) is cos θ = \(\frac{a_1a_2 + b_1b_2 }{\sqrt {a_1^2 + b_1^2 } \cdot \sqrt{a_2^2 + b_2^2 }}\)
 The angle between a pair of straight lines ax^{2} + 2hxy + by^{2} = 0 is θ = \(Tan^{1}\frac {2\sqrt {(h^2  ab)}}{(a + b)}\)
 By the cosine rule, in a triangle having sides of lengths, a, b, c, the angle between two sides of a triangle is equal to cos A = \(\frac{b^2 + c^2  a^2}{2bc}\).
Angle Between Two Lines In Three Dimensional Space
The angle between two lines in a threedimensional space can be calculated similarly 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.
cos θ = \(\dfrac{b_1\cdot b_2}{b_1b_2}\)
Further for two lines having the direction ratios as \((a_1, b_1, c_1)\), and \((a_2, b_2, c_2)\), the angle between the lines can be computed using the below formula.
cos θ = \(\dfrac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}\)
Also for two lines having the direction cosines as \(l_1, m_1, n_1\), and \(l_2, m_2, n_2\), the angle between the two lines can be computed using the following formula.
cos θ = \(l_1l_2 + m_1m_2 + n_1n_2\)
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Examples on Angle Between Two Lines

Example 1: Find the angle between two lines having slopes of 1, and 1/2 respectively.
Solution:
The gives slopes of the two lines are \(m_1\) = 1 and \(m_2\) = 1/2.
The formula to find the angle between the two lines is tan θ = \(\left\frac{m_1  m_2}{1 + m_1m_2}\right\).
tan θ = \(\left\frac{1  1/2}{1 + 1/2 \cdot 1}\right\)
tan θ = \(\left\frac{1/2}{3/2}\right\)
θ = \(\lefttan^{1} \frac{1}{3}\right\)
Answer: ∴ The angle between the lines is \(tan^{1} \left(\frac{1}{3}\right)\).

Example 2: Find the angle between two straight lines having the equations 3x + 4y  10 = 0, and 4x 5y + 2 = 0.
Solution:
The given two equations of the lines are 3x + 4y  10 = 0, and 4x 5y + 2 = 0.
Here we have \(a_1 = 3, b_1 = 4, a_2 = 4, b_2 = 5\)
The angle between the two lines can be calculated using the formula Tanθ =\(\left\dfrac{a_2b_1  a_1b_2}{a_1a_2 + b_1b_2}\right\).
tan θ =\(\left\frac{4 × 4  3 ×(5)}{3 × 4 + 4 ×(5)}\right\)
tan θ =\(\left\frac{16 + 15}{12  20}\right\)
tanθ =\(\left\frac{31}{8}\right\)
θ =\(tan^{1}\left\frac{31}{8}\right\)
θ =\(tan^{1}\left(\frac{31}{8}\right)\) ≈ 75.53°
Answer: ∴ The angle between the lines is 75.53°.

Example 3: Derive the condition for two lines with slopes m_{1} and m_{2} to be parallel and perpendicular using the angle between two lines formula.
Solution:
The angle between two lines with slopes m_{1} and m_{2} is:
tan θ = \(\left\frac{m_1  m_2}{1 + m_1 m_2}\right\).
When the lines are parallel:
We know that the angle between two parallel lines is 0°.
Then tan 0 = 0.
Substituting this in the above formula leads to m_{1}  m_{2} = 0 ⇒ m_{1} = m_{2}.
When the lines are perpendicular:
We know that the angle between two perpendicular lines is 90°. We know that tan 90° is not defined.
The above formula is NOT defined when 1 + m_{1} m_{2} = 0 ⇒ m_{1} m_{2} = 1.
Answer: The condition for two lines to be parallel is m_{1} = m_{2 }and the condition for the two lines to be perpendicular is m_{1} m_{2} = 1.
FAQs on Angle Between Two Lines
How Do You Find the Angle Between Two Lines?
The angle between two lines can be calculated from the slopes of the lines or from the equation of the two lines. The simplest formula to find the angle between the two lines is from the slope of the two lines. The angle between two lines with slopes \(m_1\), and \(m_2\) respectively is tan θ = \(\left\dfrac{m_1  m_2}{1 + m_1m_2}\right\).
What is the Angle Between Two Lines Formula?
The are two important formulas to find the angle between two lines in a coordinate plane.
 For two lines having slopes \(m_1\), and \(m_2\) respectively, the angle between the two lines is tan θ = \(\left\dfrac{m_1  m_2}{1 + m_1m_2}\right\).
 The second formula to find the angle between the two lines having the equations \(l_1 = a_1x + b_1y + c_1 = 0\), and \(l_2 = a_2x + b_2y + c_2 = 0\), is tan θ =\(\left\dfrac{a_2b_1  a_1b_2}{a_1a_2 + b_1b_2}\right\).
Can the Angle Between Two Lines be Negative?
No, the angle between the two lines is always positive. Also, the formula to find it always includes absolute value sign in it.
How To Find Angle Between Two Lines in ThreeDimensional Geometry?
The angle between two lines in threedimensional geometry, having the equations of the lines as \(r = a_1 + λb_1\), and \(r = a_2 + λb_2\), is cos θ = \(\left\dfrac{b_1 \cdot b_2}{b_.b_2}\right\).
Why is the Absolute Value Sign in the Angle Between Two Lines Formula?
The angle between two lines refers to the acute angle between them. The tan θ or cos θ formula which we use to find the angle between lines leads to an obtuse angle if the absolute value is NOT included. That is why, we include it.
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