Transitive Property of Parallel Lines

Instructor: David Karsner
The transitive property is one of the basic properties of mathematics that applies across different areas of mathematics from algebra to geometry. In this lesson you will see that it is a way to prove that multiple lines are parallel.

Parallel Lines of the City

If you live in a city that has a grid system for its streets you will be familiar with the concept that the streets either intersect or run parallel to each other. If 4th and 5th Avenue are parallel and 5th and 6th Avenue are parallel, can you say that 4th and 6th run parallel to each other? It turns out that you can! This lesson is about parallel lines and how they observe the transitive property.

Transitive Property

The transitive property states that if a=b and b=c then a=c. The definition uses equal signs but it can be stated in other ways. For example, if Lucy is taller than Danny and Danny is taller than Roger then Lucy must be taller than Roger.

Parallel Lines

Parallel lines are two or more lines that lie in the same plane but never intersect. Both criteria, lie in the same plane and never intersect, must be met to be considered parallel lines. Two lines that lie in different planes will never intersect but they are not parallel. (They are called skew lines.) Imagine a road running North-South and a power line running East-West across it. These lines never intersect but they don't lie in the same plane so they are not parallel.

The transitive property of parallel lines states that if line E is parallel to line F and line F is parallel to line G then line E is parallel to line G.

Examples

The Slope of Lines

If two lines are parallel, they have the same slope. If line A has a slope of 2/3 and line B is parallel to line A, then line B has a slope of 2/3. The transitive property allows you to extend this further. As long as the lines are parallel to each other, the slope will remain the same. If line A is parallel to B and B is parallel to line C, then the slope of A will be the same as the slope of C. Line C will also have a slope of 2/3

Corresponding Angles

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