I'm wondering about the relationship between Eculid's work and modern high school geometry. In "two column proofs," certain reasons are considered acceptible for steps in the proof, such as those shown below, from here.

#16. Definition of alternate interior angles – says that “If two angles are alternate interior, then they are on opposite sides of a transversal and are both on the interior to two lines (whether parallel or not).”

#17. Alternate interior angle theorem – says that “If two lines are parallel and alternate interior angles are formed, then the angles will be congruent to one another.”

#18. Converse of alternate interior angle theorem – says that “If alternate interior angles are congruent, then the lines that form them will be parallel to one another.”

It strikes me that many of the reasons considered acceptable in these kinds of proof are similar to, but not identical to, postulates, definitions etc. given in Euclid.

What I'm wondering is if there is some kind of modern standard for these kinds of reasons, that captures the essence of Euclid's reasons, but which uses modern insights and concepts to make them more accessible.

By which I mean basically, is there some kind of canonical source for what counts as valid reasons for steps in modern high school geometry proofs, or is what we have a kind of collective cultural habit that is loosely based on Euclid's work but is not in anyway standardized?

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    $\begingroup$ Frame challenge: "two column proofs" are evil, and we should teach students to write in clear mathematical English (meaning full sentences which mix English and notation, with a capital letter at the beginning and a period at the end!). "Valid" steps require that a student clearly explain their thinking, and that they reference axioms or previously proved lemmata, theorems, and so on. As each instructor is going to have a slightly different point of view, standardization (of the kind implied in the question) seems unlikely. $\endgroup$
    – Xander Henderson
    Mar 9, 2022 at 23:01
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    $\begingroup$ Do we expect to have canonical standards for what constitutes a good essay for English class, or a collective cultural habit? Why should we expect mathematics to be any different from any other humanities subject? $\endgroup$ Mar 10, 2022 at 0:04
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    $\begingroup$ @XanderHenderson Frame challenge challenge -- a two-column proof format is a teaching tool comparable to diagramming sentences: both would scarcely be found in the "real world". Also like sentence diagrams, a two-column format is a simple framework to examine the form of writing or thought which is more loosely structured. Finally, both devices are presented to soph HS students or earlier... if many math majors struggle to write unstructured proofs, how can we expect HS students to do so without clear guidance given that many cannot even write well-organized prose essays? $\endgroup$
    – Steve
    Mar 10, 2022 at 1:17
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    $\begingroup$ ... I mean, even my graduate-level statistics text has columnar sequences of chained equations, with justifications in a margin to the right. It's not a crazy format to use. $\endgroup$ Mar 10, 2022 at 1:25
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    $\begingroup$ The two-column proof format is a graphic organizer, no more evil than KWL charts or word webs. I think it works well when you are studying a symbolically heavy structure with strict axiomatic rules of formation. (Principia Mathematica is the image I have in mind here.) My gripe with how we teach high school geometry which gets caught up in proof formats is that we ignore all of the logical foundation of plane geometry and then demand that the students pretend to be mathematically rigorous with the scraps of facts we provide to them. $\endgroup$ Mar 10, 2022 at 4:12

1 Answer 1


In a word, no. There isn't even a standardized list of plane geometry axioms in the modern era. Even if two different curricula happened to start from the same axiomatic basis, there is no longer a single authority on which consequences of those axioms rise to the level of propositions or theorems. (I'm not sure there ever was.)

  • $\begingroup$ The link in this answer gives an excellent summary, as well as the reasons Euclid is no longer used. For those in the U.S. (using two-column proofs), it is likely they will be using a list based closely on the SMSG axioms. $\endgroup$ Mar 15, 2022 at 20:07
  • $\begingroup$ @ScottEberle My students receive a radically different treatment than SMSG. We start from rigid transformations (which is taught in middle school), and then define two figures as congruent if there is a series of rigid transformations that transforms one to the other. In the same way, two lines are parallel if one is a translation of the other. Everything flows out of that and a few postulates like two different lines cannot intersect in more than one point. This treatment seems to spring from Birkhoff's work, although I haven't dug deeper than this wikipedia article to track its lineage. $\endgroup$ Mar 15, 2022 at 21:04

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