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Defining Parallel Lines


Alignments to Content Standards: G-CO.A.1

Task

Alex and his friends are studying for a geometry test and one of the main topics covered is parallel lines in a plane. They each write down what they think it means for two distinct lines in a plane to be parallel:

  1. Rachel writes, ''two distinct lines are parallel when they are both perpendicular to a third line.''
  2. Alex writes, ''two distinct lines are parallel when they do not meet.''
  3. Briana writes, ''two distinct lines are parallel when they have the same slope.''

Analyze each definition, indicating if it is mathematically correct and if it has any drawbacks.

IM Commentary

The goal of this task is to critically analyze several possible definitions for parallel lines. 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 second definition is the one which Euclid adopted and it is pretty common in textbooks. The third definition has slight problems. 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).

A 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 downside to this definition is that it requires both an understanding of perpendicular lines and a notion of distance. On the positive side, this idea provides a nice link to coordinate geometry.

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. Teachers may wish to engage in a discussion on this front. Many textbooks, including many books that teachers will study in college, will make a slightly different definition of parallel lines, modifying Alex's definition:

two lines are parallel if they do not meet or if they are the same.

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. This has one interesting consequence. Suppose $\ell$ and $m$ are distinct parallel lines. Then $\ell$ is parallel to $m$ and $m$ is parallel to $\ell$. If we want the property of being parallel to be transitive, then this would mean that $\ell$ is parallel to itself. One way around this difficulty would be to put forward the transitive property but only for three distinct lines. 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.

 

A mathematical definition has at least three key properties which are investigated in this task:

  • It must be clearly and precisely stated with no ambiguity.
  • It must capture all possible situations or scenarios.
  • It should only use notions and prior knowledge which can be considered ''more basic.''

All three of these proposed definitions satisfy the first criterion listed above. Definition (c) fails on the second point while (a) and (c) fail on the third. 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.

This task is best suited for an in depth discussion in class. 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). The main mathematical practice relevant for this task is MP6, ''Attend to Precision.'' Most work with definitions requires great care in choice of language and a vitally important idea such as parallelism provides a perfect opportunity to discuss this.

Solution

  1. Rachel's definition of parallel lines is technically correct: if $\ell$ and $m$ are both perpendicular to a traverse line $t$ then $\ell$ and $m$ are parallel. This definition does, however, have some drawbacks. First, it refers to a transverse line $t$ that is not part of the given information. In other words, we need to choose or construct a transverse $t$. It would be better if we could find an ''intrinsic'' characterization of what it means for $\ell$ and $m$ to be parallel, without reference to other constructions. More seriously, this definition assumes that we know what it means for two lines to be perpendicular. While high school students know both what it means for a pair of lines to be perpendicular and what it means for them to be parallel, the definition of parallel lines does not require any knowledge of right angles and consequently it should not include any language making reference to lines being perpendicular.
  2. It is true that if $\ell$ and $m$ are lines in the plane that do not meet then they are parallel. This is perhaps the most classical and intuitive notion of parallel lines: a pair of ''railroad tracks'' that go on forever without meeting.

  3. Briana's definition has two basic flaws. First, it only applies to lines which have a slope: in particular, Briana's definition does not tell us whether or not pairs of vertical lines are parallel or whether or not a vertical line is parallel to a non-vertical line. So Briana's definition fails to cover all possible cases: this can be corrected by adding that all vertical lines are parallel and no vertical line is parallel to a non-vertical line. In addition, like Rachel's definition, it refers to auxiliary information, in this case the slope of the two lines. While it is true that lines with the same slope are parallel, it is possible to talk about lines being parallel with no reference to slope. Since slope is not needed in the definition, it should not be used except to make connections with prior student knowledge. Slope requires coordinates which is an additional structure on the plane, a structure which Euclid did not have at his disposal.

Elizabeth Young says:

almost 3 years

Would Alex's definition not include skew lines? In the solution a plane is referenced.

Cam says:

almost 3 years

Sure, though the intent was for the whole problem to be envisioned within a plane. Just to be safe, I've rewritten the stem slightly, trying to emphasize that this is a plane geometry problem.

It's surprisingly tricky to give a definition of parallel lines in 3-D, especially if you want to give one definition for both 2-D and 3-D simultaneously (and trickier still if you only want to terms as primitive as Euclid had at this point in the elements).