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Two different lines on the same plane can either intersect or not. If the lines do not intersect, then they are parallel. If the lines do intersect, they might be perpendicular. It can be useful to know how to identify parallel and perpendicular lines, which will be explored in this lesson.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Observing Parallel Lines

Consider the following applet showing two parallel lines and their equations in slope-intercept form.
Two parallel lines.
What characteristic of the lines remains unchanged when they are moved up or down?
Discussion

Parallel Lines

Two coplanar lines — lines that are on the same plane — that do not intersect are said to be parallel lines. In a diagram, triangular hatch marks are drawn on lines to denote that they are parallel. The symbol is used to algebraically denote that two lines are parallel. In the diagram, lines and are parallel.

One pair of parallel lines and one transversal
Rule

Slopes of Parallel Lines Theorem

In a coordinate plane, two distinct non-vertical lines are parallel if and only if their slopes are equal.

Parallel Lines

If and are two parallel lines and and their respective slopes, then the following statement is true.

The slope of a vertical line is not defined. Therefore, this theorem only applies to non-vertical lines. However, any two distinct vertical lines are parallel.

Proof

Since the theorem consists of a biconditional statement, the proof consists of two parts.

  1. If two distinct non-vertical lines are parallel, then their slopes are equal.
  2. If the slopes of two distinct non-vertical lines are equal, then the lines are parallel.

Part

Consider two distinct non-vertical parallel lines in a coordinate plane. Their equations can be written in slope-intercept form.
Suppose that the slopes of the lines are not the same. The system of equations formed by the equations above can be solved by using the Substitution Method.
Solve for
Since the expression is not undefined because its denominator cannot be zero. To find the value of the variable, can be substituted for in Equation (II).
The solution to the system formed by the equations was found. Since there is a solution for the system, the lines and intersect each other. However, this contradicts the fact that the lines are parallel. Therefore, the assumption that the slopes are different is false. Consequently, the slopes of the lines are equal.

Part

Now, consider two distinct non-vertical lines and that have the same slope Their equations can be written in slope-intercept form.
Since these lies are distinct, and are not equal. With this information in mind, suppose that the lines intersect. Solving the system of equations will give the point of intersection. The Substitution Method will be used again.
The obtained result contradicts the fact that and are different. Therefore, there is no point of intersection between the lines and This means that they are parallel lines.

Both directions of the biconditional statement have been proved.

Pop Quiz

Identifying Parallel Lines

Consider two lines on the same coordinate plane and their equations in slope-intercept form. Are the lines parallel?

Two lines and their equations.
Example

Finding a Parallel Line Through a Point

Write an equation in slope-intercept form of the line that passes through and is parallel to

line and point

Hint

What do parallel lines have in common?

Solution

Consider the given equation of the given line.
Parallel lines have the same slope. Therefore, the slope of all lines parallel to the given line is A general equation in slope-intercept form for these lines can be written.
The desired parallel line passes through By substituting the coordinates of this point into the above equation for and the intercept of the parallel line can be determined.
Solve for
Knowing the intercept, the equation of the line parallel to through can be written.
The solution can be verified by graphing the line on the same coordinate plane.
lines and point

As seen above, the graph of passes through and is parallel to the graph of

Example

Bike Path

Kevin's class is entering a contest to redesign parts of their town. They are using new software to create a model of their ideas for changes to the park. Kevin wants to propose a new bike path in the park. The bike path has to be parallel to an already existing pavement. He visualized the situation on the following graph.
A point can be seen on the graph. Kevin wants the bike path to pass through the point, as his measurements show that this is the most convenient option. What is the equation of the line, written in slope-intercept form, that corresponds to the bike path?

Hint

Find the slope of the line shown on the graph.

Solution

Two parallel lines have the same slope. To find the equation of the line that corresponds to the bike path, the slope of the line that represents the pavement has to be determined first. This can be done by using the Slope Formula.
In the above formula, and are two points on the line. Therefore, two points on the line corresponding to the pavement have to be identified.
Now and will be substituted for and respectively.
The slope of the pavement is As mentioned before, parallel lines have the same slope. Therefore, all lines parallel to the line that corresponds to the pavement will have a slope of A general equation in slope-intercept form for these lines can be written.
The line that represents the bike path passes through By substituting the coordinates of this point into the above equation for and the intercept of the parallel line can be determined.
Solve for
Knowing the intercept, the equation of the line parallel to the pavement through can be written.
The solution will be verified by plotting the line on Kevin's graph.

As seen above, the graph of is parallel to the pavement and passes through

Example

Starry Scene

Zosia wants to propose a new mural to be painted on the side of the planetarium. She starts with a moon and two stars that are already painted on the building.

Zosia wants to place more stars in the line that connects the two existing stars. She also wants to make a second line of stars that is parallel to the first and passes through the moon. The two stars and the moon can be represented on a coordinate plane.

Write the equation of the line of stars that passes through the moon. Give the answer in slope-intercept form.

Hint

Find the slope of the line that passes through the stars.

Solution

The coordinates of the points that represent the stars are and Since two points on the line are known, the line's slope can be found using the Slope Formula.
Simplify right-hand side
The slope of the line that passes through the stars is or Parallel lines have the same slope. Therefore, all lines that are parallel to the line that passes through the stars will have a slope of A general equation in slope-intercept form for these lines can be written.
The parallel line should pass through the moon, which in the coordinate plane has coordinates These two numbers will be substituted into the above equation. Then, the intercept of the parallel line can be determined.
Solve for
Now, the equation of the parallel line can be completed.
The solution will be verified by graphing the above line on the visual representation of Zosia's proposed mural.
Explore

Observing Perpendicular Lines

The following applet shows two perpendicular lines and their equations in slope-intercept form.
Two perpendicular lines.
The lines remain perpendicular as they are moved. What characteristic of the lines does not change? What is the relation between the slopes of these two perpendicular lines?
Discussion

Perpendicular Lines

Two coplanar lines — lines that are on the same plane — that intersect at a right angle are said to be perpendicular lines. The symbol is used to algebraically denote that two lines are perpendicular. In the diagram, lines and are perpendicular.

Two lines intersecting at a right angle
Rule

Slopes of Perpendicular Lines Theorem

In a coordinate plane, two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals.

Perpendicular Lines

If and are two perpendicular lines and and their respective slopes, the following relation holds true.

This theorem does not apply to vertical lines because their slope is undefined. However, vertical lines are always perpendicular to horizontal lines.

Proof

Since the theorem is a biconditional statement, the proof consists of two parts.

  1. If two non-vertical lines are perpendicular, then the product of their slopes is
  2. If the product of the slopes of two non-vertical lines is then the lines are perpendicular.

Part

Let and be two perpendicular lines. Therefore, they intersect at one point. For simplicity, the lines will be translated so that the point of intersection is the origin.
Let and be the slopes of the lines and respectively. Next, consider the vertical line This line intersects both and
Since and are assumed to be perpendicular, is a right triangle. Using the Distance Formula, the lengths of the sides of this triangle can be found.
Side Points
Length
Since is a right triangle, its side lengths satisfy the Pythagorean Equation.
The next step is to substitute the lengths shown in the table.
Simplify
It has been proven that if two lines are perpendicular, then the product of their slopes is

Part

Here it is assumed that the slopes of two lines and are opposite reciprocals.
Consider the steps taken in Part This time, it should be found that is a right triangle.
If the lengths of the sides of satisfy the Pythagorean Theorem, then the triangle is a right triangle.
The side lengths, which were previously found in Part can be substituted into the above equation.