I find the interior angle sum of the triangle fascinating, because it’s such a basic concept, and still alien to a lot of “not math” people. In my experience, which I admit is limited, it is often introduced, at least these days, by exploration. You cut out a random triangle, rip it up, and reassemble the angles. Wow, it’s a straight line! You do it again, or look at the many random triangles your classmates have done it and it’s pretty close to a straight line for all of them. And then you don’t deal with it again, or you are presented with a formal proof.
And the formal proofs for triangle are pretty straight forward. You can make them seem alien to non-math people by being overly formal and not reiterate the obviousness of the underlying relationships, but even then they require “thinking outside the triangle”.
I recently came up with a different approach, and I have to questions about it.
Is it well known and I’m just not googling right? (I suspect that is the case.)
What separates it from being good enough for Euclid?
Draw a triangle ABC (labeled anti-clockwise). Draw a line segment along AC starting outside the triangle, passing through A and ending inside it. Mark one end of the line segment. (You could make it an arrow, but I think that creates an idea it should move in the direction of the arrow.)
Rotate the segment clockwise around A until it matches up with AB. Shift it along AB to B.
Rotate it clockwise around B until it matches up with BC. Shift it along BC to C.
Rotate it clockwise around C until it matches up with AC.
It’s now rotated in the same direction for each of the interior angles, and has obviously made a half rotation in total. And it’s obvious that it would work for any triangle.
I’m sure that to make it mathematically rigorous this “proof” would require more assumptions than Euclid, but I think it could be a better introduction to how mathematics works for a lot of kids.
Also it would more firmly establish the “half circle” as the more basic entity for angles and poison kids against the Tau propagandists at an early age.
Keep in mind: the sum of the interior angles of a hyperbolic triangle is strictly less than a straight angle. So, whatever visual trick you use, you must make it clear where you apply the parallel postulate.
You are not, and that is the problem. Your procedure
applies to a hyperbolic triangle, but
in that case the total rotation at the angles is strictly less than a half-rotation, and you need explicitly to account for the rotation given by parallel transport. What I mean is, your procedure does not seem to prove anything about the sum of the angles.
For kids learning about the interior angle sum for the first time I think it’s best not to introduce non-euclidian geometries. For someone at a higher level enjoying a different visualization, how do I apply the parallel postulate? Just preface my explanation with this?
We presume a space where Euclid’s postulates hold, including:
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
As @DPRK suggests the problem is the possibility of parallel transport.
To be honest the simplest proof seems to me that visually compellimg fact that when a line meets two parallel lines, the alternate interior angles are equal. Of course, that requires drawing a line though a vertex that is parallel to the side opposite that vertex and requires the parallel postulate.
You could use “Playfair’s Axiom”: given a line in the plane and a point not on the line, there exists a unique line through the point parallel to the given line.
That will enable you to “transport” angles as in Proposition 29 mentioned by @Hari_Seldon , and of course Proposition 32.
I don’t disagree that that is a pretty good and a fairly simple proof. And that it is a proper mathematical proof. It requires a kid to understand the mathematical concept of parallel though. I can imagine an annoying kid going:
Why are the angles equal?
Because the two lines are parallel.
Why does that make them equal?
…
I think this visual examination goes well along with the cutting out triangles and assembling the corners, giving even kids who dislike math a taste of what it means to prove something.
Both of them nicely link the interior angle sum to the straight angle, which is less abstract for a kid than “180 degrees”.
That would not be the kid being annoying, though; that’s Proposition 29, which must be proved using whichever version of the Parallel Postulate you find most palatable, or more comprehensible to kids considering we might ask them to imagine the notion of a pair of straight lines that go on to infinity but never meet.
It would be annoying if I was trying to reach kids, including that one, with an approach that only required them to accept a small number of intuitively true things. It’d be “correct”, but still true.
I find it much less likely that there would be a kid in that class whose brain worked in non-euclidian space and requested proof that translating the “test segment” wouldn’t influence rotation.
