Parallel and perpendicular lines are sets of lines which display a specific relationship between them. Parallel lines are lines that do not intersect and the perpendicular lines intersect at right angles. The following diagram shows the locations of the houses of three friends.

Parallel and Perpendicular Lines


It can be seen that Cindy's and Rudy's houses are situated in parallel streets, while Nancy's house is on a street which is perpendicular to both these streets. This illustrates how the relationship of two lines to a third line can lead to the relationship between the two lines.
1. In a plane if two lines are perpendicular to the same line, then they are parallel to each other.
2. If two lines are parallel to the same line, then they are parallel to each other.

Parallel lines are coplanar lines that have no points in common or have all points in common.
The parallel relationship is denoted by the symbol ||.In the first case, when the two lines have no points in common, we have two distinct and parallel lines. In the diagram shown below lines 'l' and 'm' are parallel lines with no common point.

Parallel Lines

If two lines under consideration have all points in common, then two lines coincide.

Parallel lines are necessarily coplanar. Non coplanar lines in space without common points are called skew lines.
Perpendicular lines are intersecting lines and they intersect to form four right angles at the point of intersection. 

Perpendicular Lines

Perpendicular lines are the most common lines that we see around us in real life. The walls of your room are perpendicular to the floor.

From a point that lies in the exterior of a line one and only one perpendicular line can be drawn.
The adjacent sides of a rectangle or a square are perpendicular to each other.
The diagonals of a rhombus, a square and a kite are perpendicular lines.
The postulates or theorems related to the properties of parallel lines are related to the angles made by a transversal which intersect them.

 If two parallel lines are cut by a transversal, then
Theorem / Postulate
Explanation
Diagram
according to Corresponding angles Postulate
each pair of corresponding angles is
congruent.
∠1 ≅ ∠ 3
∠2 ≅ ∠ 4
∠ 5 ≅ ∠ 7
∠ 6 ≅ ∠ 8
 Parallel Lines Angles
according to Alternate Interior angles
theorem, each pair of alternate interior angles
is congruent.   
∠ 2 ≅ ∠ 6
∠ 3 ≅ ∠ 7
according to same side interior angles
theorem each pair of same side interior
angles is supplement.
m ∠2 + m ∠3 = 180º
m ∠6 + m ∠7 = 180º
according to alternate exterior angles
theorem each pair of alternate exterior
angles is congruent.
∠ 1 ≅ ∠ 5
∠ 4 ≅ ∠ 8

The above angle properties are used in proving many geometrical problems. These properties are used to check whether two lines are parallel or not.

Properties of perpendicular lines

1. In a plane if two lines are perpendicular to the same line, then they are parallel to each other.
2. If two parallel lines are cut by a transversal which is perpendicular to the lines, then each pair of same side interior angles is also congruent.

The parallel lines are identified by the congruent corresponding angles or by congruent alternate interior angles or by congruent alternate exterior angles or supplement same side interior angles.

A perpendicular line in a diagram is identified by the right angle marked using box like ⌈ or ⌉ signs. If the linear pair of angles formed are congruent, then the intersecting lines are perpendicular to each other.
The slopes of parallel lines are equal.

Example:
Consider the lines represented by equations y = 2x - 3  and y = 2x + 5. Both the equations are given in y = mx + b form
Hence the slopes of both the lines m = 2. Thus the lines represented by these two equations are parallel.

Parallel Lines Slope

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line.
If m1 and m2 are the slopes of two perpendicular lines, then m1 x m2 = -1.

Example:
Consider the lines represented by equations y = 2x - 3 and y = -`1/2`x + 5.  Both the equations are given in y = mx + b form and the slopes of the lines are m1= 2 and m2 = -`1/2`.
m x m = 2 x (-`1/2`) = -1.
Thus the two given equations represent perpendicular lines.

Perpendicular Lines Slope

Following table contrasts some of the properties of parallel and perpendicular lines.

Parallel lines
Perpendicular lines
1. Parallel lines do not intersect.
1. Perpendicular lines intersect at right angles.
2. The slopes of two parallel lines are equal.
2. The slopes of two perpendicular lines are negative
reciprocals of each other.
3. For two parallel lines,a line perpendicular
to one of the lines is also perpendicular to
the other line, Similarly a line parallel to one
of the lines is also parallel to the other line.
3. Given two perpendicular lines, then a line
perpendicular to one of the lines is parallel to the
other line. Similarly a line parallel to one of the lines
is perpendicular to the other line.

The equations of two parallel lines differ only in the constant term or the y intercept, 2x + 5y = 7 and 2x + 5y = -9 represent the equations of two parallel lines.

Similarly if the equation of a line is y = 3x -4, then the equation of a line parallel to it can be assumed as
y = 3x + k and the value of k can be found from some other given conditions.

If the equation of a given line is ax + by =c, then the equation of a line perpendicular to this line can be assumed as bx - ay = k or -bx + ay = k. (Interchange the coefficients of x and y and change the sign of one of them and also the constant term).

For example, the lines 2x - 7y = 11 and -2x + 7y = 9 are the equations of two possible lines which are perpendicular to
7x + 2y = 3.
We use the following facts in graphing parallel and perpendicular lines of a given line.
  1. The slopes of parallel lines are equal.
  2. The product of slopes of two perpendicular lines = -1.

Solved Example

Question: Given the line 2x + y = 5. and a point (-3,4)

    Graph the line parallel to the given line and passes through the given point.
    Graph the line perpendicular to the given line and passes through the given point.
Solution:
 
Let us first find the slope of the given line 2x + y = 5.
y = -2x + 5                         Equation written in slope intercept form y = mx +b
Slope of the given line = -2
Slope of the parallel line = -2
Slope of the perpendicular line = `1/2`.

We first mark the given point (-3,4).

We can use the given point and find the other point using the slope.
Slope = -rise / run.
Hence for the parallel line to find the second point we can take rise = -2 and run 1. This means we move 2 units down and 1 unit to the right to reach the second point as (-3 + 1, 4 -2) = (-2, 2).
Now we join the two points (-3, 4) and (-2, 2) to get the line parallel to the given line.

Graph Parallel Lines

Now for the perpendicular lines, the slope = `1/2`.
Hence we can consider the rise as 1 and run as 2.
We move 1 unit up and 2 units to the right to get the second point which is ( -3 + 2, 4 +1) = (-1, 5).

Graph Perpendicular Line