Parallel lines are straight lines that extend to infinity without touching at any point. Perpendicular lines cross each other at a 90-degree angle. Both sets of lines are important for many geometric proofs, so it is important to recognize them graphically and algebraically. You must know the structure of a straight-line equation before you can write equations for parallel or perpendicular lines. The standard form of the equation is "y = mx + b," in which "m" is the slope of the line and "b" is the point where the line crosses the y-axis.

### Parallel Lines

Write the equation for the first line and identify the slope and y-intercept.

Example: y = 4x + 3 m = slope = 4 b = y-intercept = 3

Copy the first half of the equation for the parallel line. A line is parallel to another if their slopes are identical.

Example: Original line: y = 4x + 3 Parallel line: y = 4x

Choose a y-intercept different from the original line. Regardless of the magnitude of the new y-intercept, as long as the slope is identical, the two lines will be parallel.

Example: Original line: y = 4x + 3 Parallel line 1: y = 4x + 7 Parallel line 2: y = 4x - 6 Parallel line 3: y = 4x + 15,328.35

### Perpendicular Lines

Write the equation for the first line and identify the slope and y-intercept, as with the parallel lines.

Example: y = 4x + 3 m = slope = 4 b = y-intercept = 3

Transform for the "x" and "y" variable. The angle of rotation is 90 degrees because a perpendicular line intersects the original line at 90 degrees.

Example: x' = x_cos(90) - y_sin(90) y' = x_sin(90) + y_cos(90)

x' = -y y' = x

Substitute "y'" and "x'" for "x" and "y" and then write the equation in standard form.

Example: Original line: y = 4x + 3 Substitute: -x' = 4y' + 3 Standard form: y' = -(1/4)*x - 3/4

The original line, y = 4x + b, is perpendicular to new line, y' = -(1/4)_x - 3/4, and any line parallel to the new line, such as y' = -(1/4)_x - 10.

#### Tip

For three-dimensional lines, the process is the same but the calculations are much more complex. A study of Euler angles will help understand three-dimensional transformations.