In Fig . (ii), we have ∠ MPQ = ° – θ . Therefore, θ = ° – ∠ MPQ. Now, slope of the line l m = tan θ = tan ( ° – ∠ MPQ) = – tan ∠ MPQ MQ MP = − y .
Consequently, we see that in both the cases the slope m of the line through the points ( x , y ) and ( x , y ) is given by m . . Conditions for parallelism and perpendicularity of lines in terms of their slopes In a coordinate plane, suppose that non-vertical lines l and l have slopes m and m , respectively. Let their inclinations be α and β , respectively.
If the line l is parallel to l (Fig . ), then their inclinations are equal, i.e., α = β , and hence, tan α = tan β Therefore m = m , i.e., their slopes are equal. Conversely, if the slope of two lines l and l is same, i.e., m = m . Then tan α = tan β .
By the property of tangent function (between ° and °), α = β. Therefore, the lines are parallel. Fig . (ii) Fig .
STRAIGHT LINES Hence, two non vertical lines l and l are parallel if and only if their slopes are equal. If the lines l and l are perpendicular (Fig . ), then β = α + ° . Therefore,tan β = tan ( α + °) = – cot α = tan α i.e., m = m or m m = – Conversely, if m m = – , i.e., tan α tan β = – .
Then tan α = – cot β = tan ( β + °) or tan ( β – °) Therefore, α and β differ by °. Thus, lines l and l are perpendicular to each other. Hence, two non-vertical