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Parallel and Perpendicular Lines

Proving Relationships of Parallel and Perpendicular Lines

Parallel and perpendicular lines can be analyzed in the coordinate plane. Here, their slopes are used to prove their inherent characteristics.
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

Let and be parallel lines, and and be 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. Any two distinct vertical lines are parallel.

Proof

Since the theorem consists of a biconditional statement, the proof will consists in 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 1

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 different. This means that and are not equal. Using the Substitution Method, the system of equations formed by the equations above can be solved.
Solve for
Since the expression is not undefined because its denominator is different than zero. To find the coordinate of the solution, can be substituted for in Equation (II).
The solution to the system formed by the equations was found. This implies that the lines and intersect. 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 2

Now, consider two distinct non-vertical lines and that have the same slope Their equations can be written in the slope-intercept form. Since these are distinct lines, 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. Again, the Substitution Method will be used.
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.

Proof

Slopes of Parallel Lines Theorem

Consider two lines in a coordinate plane. In slope-intercept form, their equations can be written as follows. The intersection of these lines is the - point that satisfies both equations. Setting the equations equal to each other makes it possible to solve for the -coordinate of the intersection point.
The -coordinate of the point of intersection is However, if the lines are parallel, by definition, there will be no point of intersection. Thus, must be undefined, which occurs when the denominator is zero. Therefore, the slopes of parallel lines are equal.
The above proof can be summarized in a flowchart.
Rule

Slopes of Perpendicular Lines Theorem

In a coordinate plane two nonvertical lines are perpendicular if and only if the product of their slopes is

This can be proven using right triangles.

Proof

Slopes of Perpendicular Lines Theorem


Consider the perpendicular lines and in the coordinate plane, that form a right triangle with the -axis.

Using the points on each line, the slopes of and can be written as follows. Multiplying the slopes gives their product. Notice that the product equals if and only if the following is true. The Geometric Mean Altitude Theorem states that the altidtude of a right triangle is equal to the geometric mean of the segments the altitude creates with the hypotenuse. Notice that is the altitude of the right triangle in the diagram. Also, divides the hypotenuse into the segments and Thus, is a true statement. Therefore, the product of the slopes of two perpendicular lines equals

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Exercise

The points and are vertices of a quadrilateral. Prove that quadrilateral is a rectangle.

Show Solution
Solution

To begin, we can graph the quadrilateral in a coordinate plane.

If quadrilateral is a rectangle, both pairs of opposite sides will be parallel and all pairs of adjacent sides will be perpendicular. Thus, we must show that

  • ,
  • , and
Note that, if the first two criteria is proven true, it is sufficient to show that one pair of adjacent sides is perpendicular. First, we can determine the slopes of each side using the coordinates of the points and the slope formula. Then we can compare the slopes to see how the sides relate. We'll start with
Thus, has the slope The slopes of the other line segments are calculated in the same way.
Line segment Slope formula Simplified Slope

From the slopes above, we can see that and have equal slopes. Thus, they are parallel. The same can be said for and To determine if we can see if the product of their slopes equals Thus, It follows that all pairs of adjacent sides are perpendicular. Therefore, the quadrialteral is a rectangle.

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