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# Angles between two lines

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 Sub Topics The Angles between two lines is defined as shown below: If the lines are parallel then the angles between both lines is zero. Suppose we have angle ‘∅’in between two lines then it always satisfy the inequalities condition. => 0 $\leq$ ∅ $\leq$ $\frac{\pi}{2}$ If the Slope of both the lines is a1 and a2 then the angle ∅ is obtained from: Then we can write it as: => Tan ∅= | $\frac{a_{1} - a_{2}}{1+a_{1} a_{2}}$ | When we find the value of a1 a2 by above formula then we get: =>1 + a1 a2 = 0; => a1 a2 = -1; This condition follows when the lines are perpendicular and also ∅ equal to $\frac{\pi}{2}$. Out of both lines, one line is parallel to y – axis then there is no Slope and the angle ∅ is deduced by using the other line. Suppose we have one slope equals to zero then the angle between both the lines is the angle between the x – axis and one of the lines. Let slope of one line is zero and slope of other line is ‘a’, then we get the formula which is given below: => Tan ∅ = |a|; Suppose one of the slope is infinite and that line is parallel to the y – axis, then the angles between both the lines is same as the angle between one line (which has slope ‘a’) and y – axis. So the formula for this statement is given below: => Tan ∅ = |1 / m|, We get this formula when one of the slope a1 a2 approaches to ∞. We find the angle between two lines by using vector product method by using formula which is given below: Cos ∅ = | $\frac{\vec{m} . \vec{n}}{(\vec{m}) (\vec{n})}$ | Where ‘m’ and ‘n’ are two vectors and ∅ is angle between two lines.

## Angle between two line formula

The Intersection between two lines defined as an Angle between two lines. Here we will see how to find the angle between two lines. The formula is given by:
⇒ tan∅ = tan (a2 – a1),
= (tan a2 - tan a1 )/ (1 + tan a1 tan a2 )
To find the angle between two lines we have to follow the steps explained below:
Step1: To find the angle first of all we have to assume the angle between two lines.
Step2: Then find the value of slopes of the given lines. Let the line be ‘A1’ and ‘A2’.
Step3: Now determine the Tangent difference between two angles using trigonometric Functions.
Step4: Using these steps we can easily get the angle between two lines.
We know that,
a2 = a1 + ∅,
∅ = a2 – a1,
Now, calculate the value of slopes of line ‘A1’ and ‘A2’.
Using trigonometric Functions:
tan∅ = tan (a2 – a1),
=(tan a2 - tan a1 )/ (1 + tan a1 tan a2 )
If tan a1 = m1 (m1 is the Slope of the line A1)
⇒ tan a2 = m2 (m2 is the Slope of the line A2)
So, value of ‘m2’ is greater than ‘m1’, (m2 > m1).
Substitute these values in the given equation and we get:
Tan ∅ = (m2 -m1)/ ( 1 + m1 m2 )
Using this formula we find angles between two lines.
Suppose the value of m1 = ½ and m2 = 4 t and we have to find the angle between two lines m1 and m2?
We know that the value of m2 is always equal to the value of m1.
Given, m1 = ½ and m2 = 4, the formula to find the angle between two lines is given as:
tan∅ = (m2 -m1)/ ( 1 + m1 m2 )
Plug in value of m1 and m2 in the given formula:
Tan ∅ = {4-(1/2)}/(1 + ½ * 4),
={(8-1)/2}/(1 + 2),
tan∅ = 7/6.
This is all about the angle between two line formula and how to use it while solving problems related to it.