3/6/2023 0 Comments Second the notion definition![]() The teacher may wish to tell students that there are problems with all three definitions (though the problem with the second one is not that it is incorrect but rather that it is not a good way to define what it means for two lines to be parallel). This task is best suited for an in depth discussion in class. The teacher should make sure that students correct (c) and see that, while (a) is technically correct, it introduces an auxiliary construction and relies on knowledge of what it means for two lines to be perpendicular. Definition (c) fails on the second point while (a) and (c) fail on the third. It should only use notions and prior knowledge which can be considered ''more basic.''Īll three of these proposed definitions satisfy the first criterion listed above.It must capture all possible situations or scenarios.It must be clearly and precisely stated with no ambiguity.Either of these definitions is suitable: the fact that both are in use provides a good example why it is critical to communicate clearly when we reason mathematically.Ī mathematical definition has at least three key properties which are investigated in this task: One way around this difficulty would be to put forward the transitive property but only for three distinct lines. If we want the property of being parallel to be transitive, then this would mean that $\ell$ is parallel to itself. Then $\ell$ is parallel to $m$ and $m$ is parallel to $\ell$. ![]() ![]() Suppose $\ell$ and $m$ are distinct parallel lines. ![]() To briefly expand on this idea, we note that if we drop the distinctness requirement in Alex's definition, a line $\ell$ is not parallel to itself. Two lines are parallel if they do not meet or if they are the same. Many textbooks, including many books that teachers will study in college, will make a slightly different definition of parallel lines, modifying Alex's definition: Teachers may wish to engage in a discussion on this front. We note that the task deliberately avoids the issue of whether a line is considered to be parallel to itself, by assuming the lines to be distinct. On the positive side, this idea provides a nice link to coordinate geometry. The downside to this definition is that it requires both an understanding of perpendicular lines and a notion of distance. It is important that students should know all of these different ways in which the notion of parallel lines connect to other ideas that they have studied such as slope and right angles (even if these other ideas do not enter into the definition of parallel lines).Ī fourth idea for defining parallel lines, closely related to Rachel's idea, is relatively common and the teacher may wish to discuss this as well: two lines are parallel when they are ''everywhere equidistant.'' This means that if a perpendicular is drawn at any point $P$ on one of the two lines, then it will meet the second line at a point $Q$ and the distance $|PQ|$ does not depend on the chosen point $P$. The third definition has slight problems. The second definition is the one which Euclid adopted and it is pretty common in textbooks. The first definition is mathematically sound but it is also the most awkward as it describes a critical property of parallel lines and is most appropriately seen as a theorem rather than a definition. The goal of this task is to critically analyze several possible definitions for parallel lines.
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