Question

How will you Prove Mathematically:

In case of Parallel Lines, the Slopes of the two are equal that is : \(\color{green}{m_1 = m_2}\)

And in case of Perpendicular Lines, the Slopes of the two are negative reciprocals of one another or you can say their Product is \(-1\) that is : \(\color{blue}{m_1m_2 = -1}\)..

Answer 1

Slope of a line is also the tangent of the angle it makes with the x- axis
here \[\tan \theta_1=m_1\]
and
\[\tan \theta_2=m_2\]
If the two lines are parallel, then they will make equal angle with x-axis,
\[\theta_1=\theta_2\]
taking tangent both sides
\[\tan\theta_1=\tan \theta_2\]
so
\[m_1=m_2\]

Answer 2

Isn't the proof graphical??

Answer 3

Nope, it's analytical
I just made the figure to show the angles

Answer 4

Is there difference between analytical and Mathematical??

Answer 5

Actually I have one formula from which I can prove the above..

Answer 6

Here both are the same, :)
I think you could use the formula

Answer 7

By mathematical we mean using formula, no???

Answer 8

I guess

Answer 9

The formula says:

The angle between the two lines is given by:

\[\large \color{blue}{\tan(\theta) = \frac{|m_1-m_2|}{1 + m_1 m_2}}\]