The measure of steepness and direction of a straight line is given by its slope. The slope is usually represented by the letter m. In the given figure, if the angle of inclination of the given line with the x-axis is θ then, the slope of the line is given by tan θ.

The slope of a line is given as m = tan θ. If two points A \((x_1,y_1)\) and B \((x_2,y_2) \) lie on the line with (\( x_1\) ≠ \(x_2\)) then the slope of the line AB is given as:

m = tan θ = \( \frac{y_2~-~y_1}{x_2~-~x_1}\)

Where θ is the angle which the line AB makes with the positive direction of the x-axis. θ lies between 0° and 180°

It must be noted that θ = 90° is only possible when the line is parallel to y-axis i.e. at \( x_1 \) = \( x_2,\) at this particular angle the slope of the line is undefined.

Conditions for perpendicularity, parallelism, and collinearity of straight lines

**For parallel lines:**

Consider two parallel lines given by \( l_1\) and \( l_2 \) with inclinations α,β . For two lines to be parallel their inclination must also be equal i.e. α=β. This results in the fact that tan α = tan β. Hence, the condition for two lines with inclinations α,β to be parallel is tan α = tan β.

Therefore, if the slopes of two lines on the Cartesian plane are equal then the lines are parallel to each other.

Thus, if two lines are parallel then, \( m_1\) = \( m_2 \) .

Generalizing this,n lines then they are parallel only when the slopes of all the lines are equal.

If the equation of the two lines are given as ax + by + c = 0 and a’ x + b’ y + c’=0, then they are parallel when ab’ = a’b. (How? You can arrive at this result if you find the slopes of each line and equate them.)

**For perpendicular lines:**

In the figure, we have two lines \( l_1\) and \( l_2 \) with inclinations α,β. If they are perpendicular, we can say that β = α + 90°. (Using properties of angles)

Their slopes can be given as:

m1 = tan(α+90°) and \( m_2 \) = \( tan~α\).

⇒\( m_1 \) = – cot α = \( -~ \frac{1}{tan~\alpha}\) = \( -~\frac {1}{m_2}\)

⇒\( m_1 \) = \( -\frac {1}{m_2} \)

⇒\( m_1 ~×~m_2 \) = -1

Thus, for two lines to be perpendicular the product of their slope must be equal to -1.

If the equation of the two lines is given by ax + by + c = 0 and a’ x + b’ y + c’ = 0, then they are perpendicular if, aa’+ bb’ = 0. (Again, you can arrive at this result if you find the slopes of each line and equate their product to -1.)

**For collinearity:**

For two lines AB and BC to be collinear the slope of both the lines must be equal and there should be at least one common point through which they should pass. Thus for three points A, B, and C to be collinear the slopes of AB and BC must be equal.

If the equation of the two lines is given by ax + by + c = 0 and a’ x+b’ y+c’ = 0, then they are collinear when ab’ c’ = a’ b’ c = a’c’b.

**Angle between two lines:**

When two lines intersect at a point then the angle between them can be expressed in terms of their slopes and is given by the following formula:

tan θ = |\( \frac{ m_2~-~m_1}{1~+~m_1~ m_2}\)| , where \( m_1 ~and~ m_2\) are the slopes of the line AB and CD respectively.

If \( \frac{ m_2~-~m_1}{1~+~m_1~ m_2}\)is positive then the angle between the lines is acute. If \( \frac{ m_2~-~m_1}{1~+~m_1~ m_2}\) is negative then the angle between the lines is obtuse.

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