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In Euclid's The Elements, Book 1, Proposition 4, he makes the assumption that one can create an angle between two lines and then construct the same angle from two different lines. I do not see anywhere in the list of definitions, common notions, or postulates that allows for this assumption. When teaching my students this, I do teach them congruent angle construction (with straight edge and compass), but do not understand why Euclid is allowed to make this logical leap. Thanks for any help you can give.

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    $\begingroup$ If I'm not mistaken, this is exactly one of the issues that full axiom systems needed to correct - this sort of rigid angle motion is a "hidden assumption". Dunham talks about it in "Journey Through Genius" and no doubt in Hilbert's geometry you can find more about it, but I couldn't find a really authoritative web resource other than the always-helpful David Joyce Euclid site: mathcs.clarku.edu/~djoyce/java/elements/bookI/propI4.html $\endgroup$ – kcrisman Feb 1 at 4:53
  • $\begingroup$ That said, this isn't really a math education question and perhaps belong more on math.stackexchange. $\endgroup$ – kcrisman Feb 1 at 4:54
  • $\begingroup$ In Prop. I.4, two triangles are given and no angles are constructed. In Prop. I.23, the construction of a congruent angle is demonstrated. I'm not sure whether you meant I.4 or I.23 (or some other proposition in which an angle is constructed). $\endgroup$ – user1027 Feb 2 at 3:22
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    $\begingroup$ @kcrisman I personally voted against the close because I think the thrust of the question was "How through history have geometry teachers addressed the hand-waviness at the start of Elements?" At any rate, it seemed more than possible to answer this quetsion as an educator more than a mathematician, so I would argue that this question is either fine here or can be easily edited to direct the conversation in a way that should satisfy everyone. $\endgroup$ – Matthew Daly Feb 2 at 6:45
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    $\begingroup$ @MatthewDaly Sure, I can buy that point of view as well, and your answer is a fine one. I just didn't see anything that indicated it was asking about that point of view and it seemed more like a more standard math or perhaps math history question. $\endgroup$ – kcrisman Feb 4 at 3:07
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I have found this site to be an invaluable online resource for commentary. Long story short, nobody in history ever wrote an authoritative second edition of Elements, so we're still "stuck" with Euclid's original thoughts. Undeniably a work of genius that deserves its place in history, but it does have a few rough patches.

Here are David Joyce's thoughts on I.4. It's a bit of a hot mess. I personally think SAS should have been taken as the definition of congruence, another term that Euclid fails to define. He talks about superposition, which one should probably interpret like our current notion of a series of rigid motions. So it evidently isn't an angle construction so much as "sliding" $\triangle ABC$ such that $\overline{AB}$ lines up with $\overline{DE}$ and $\angle BAC$ lines up with $\angle EDF$ and "proving by intimidation" that $C$ must fall on $F$ because where else could it be?

It's not his best proof by any stretch. But I like to remind my students that it's impossible to build a dictionary where every word is defined in terms of previous words in the dictionary, and it's even difficult to imagine trying to do such a thing. Given that mindset and the fact that Euclid gets up to full speed as quickly as he does, I find myself able to "forgive" some of the early proofs like this.

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    $\begingroup$ There is a second edition to The Elements, after a fashion. It's David Hilbert's Foundations of Geometry (1899). $\endgroup$ – G. Allen Jan 30 at 0:30
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Let the two original lines be $AB$, $AC$, with the angle to recreate as $\angle BAC$.

Draw a line parallel to $AC$ that passes through $AB$ and is not $AC$. Call this line $DE$.

Where $DE$ and $AB$ intersect ($F$), drop the perpendicular of $DE$ to $AC$, and call this point $G$.

$AG$ and $GF$ can be used to recreate the angle.

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