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Perpendicular Lines

A line is a never ending locus of points on a plane at both the ends. When two lines intersect at a point they make an angle. The angle that measure exactly 90 degrees is a right angle. Geometry is based upon lines and line segments. Perpendicular lines are everywhere in geometry. These lines always have a common distance between themselves at any two corresponding points. If we rotate any one of the perpendicular lines by an angle of 90 degrees we easily get parallel lines. 
There always exist an equal perpendicular distance at all points in between parallel lines.

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Definition

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Perpendicular lines can be defined as the following two ways.

1) Two intersecting lines that form right angle (angle of 90$^{\circ}$) at their intersection point, are known as perpendicular lines or lines perpendicular to each other. These are shown in the following figure -
What is Perpendicular Line
There are four right angles formed at the intersection points. We can also say that if all four angles formed at the point of intersection of two lines are equal to one another, then the lines are said to be perpendicular lines.

2) If a line meets with another and the adjacent angles formed at the point where they meet, are equal then we can say that the lines are perpendicular to each other, as shown in the following figure -
Perpendicular Lines
In this case, one lines constructs a linear pair with another line. So, if both angles are equal, then eventually both angles will be 90$^\circ$ each.

Slope of Two Perpendicular Lines
If two lines are perpendicular to each other, then product of the slopes of both lines is -1. More specifically, if the slope of one line is $m_{1}$ and that of another is $m_{2}$, then the two lines will be perpendicular to each other when -
$m_{1}\ .\ m_{2}=-1$
There are many applications of the above relation of slopes of perpendicular lines. It is quite useful in generating equations of lines that are perpendicular and also in calculating length of perpendicular drawn from a point to a line.

Suppose we are given a line with slope ‘m’. Then the slope of the line perpendicular to it will be -$\frac{1}{m}$, and the slope of the line parallel to it will be ‘m’ only. This result can be used easily to find the slope of line perpendicular to another line whose slope is give. 


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