Angles Between Two Lines

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.
=>
                 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 ∅= 
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 .
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 ∅ =
Where ‘m’ and ‘n’ are two vectors and ∅ is angle between two lines.