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.
Since the theorem consists of a biconditional statement, the proof will consists in two parts.
Both directions of the biconditional statement have been proved.
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
The points and are vertices of a quadrilateral. Prove that quadrilateral is a rectangle.
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
|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.