The online math tutor I’m using has an exercise that at one point depends on angles being congruent based on the side-side-side triangle postulate. The problem is I can’t find where that premise was explicitly proven- that if two triangle’s sides are all congruent, then all of their angles are congruent. The latter is presumed from the fact that based on the former the two triangles are defined as congruent; except that the side-side-side postulate doesn’t seem to be that rigorously proven in the examples I’m given; the tutor does the equivalent of saying it’s intuitively obvious. Without resorting to trigonometry which is beyond the scope of what I’m currently studying, how do you demonstrate that congruent sides means congruent angles?
I didn’t pick through these, but here’s a handful of proofs.
ETA, do you have to do this? If the teacher is telling you something is a given or axiomatic, you don’t have to prove it, it’s considered true.
I mean, they could tell you for this proof, Circle A is congruent to Triangle B and you have to run with it.
It’s and axiom, so it is not explicitly proven. The same as a point with is defined as something having no dimensions.
Here is something to try to visualize :
Take any triangle, pick a side. Use one corner of the picked side as the center and draw a circle such that it passes through the apex. Repeat with the opposite corner of the picked side.
The two circles with either intersect at two points or one point (depending on the triangle) and will give you a good visualization
Do you mean three sides?
Anyways your answer is in Euclid’s Elements, Book I, Proposition 8
No, I meant two triangles having three congruent sides.
It is by no means an axiom; it is proved by Euclid as a Proposition in Book I, as mentioned above.
That’s the proposition, though. ??
Apparently Euclid himself wasn’t perfect; commentators on his Elements point out places where his logic isn’t watertight by modern standards. And of course at some point you have to say well-duh-it’s-obvious or else you couldn’t reason about anything at all. I will say that the equivalency of a triangle and a trigon is poorly established beyond the well-duh level. I’ll take that side-side-side is a given once you accept the side-angle-side proof.
P.S. Joey P, The course didn’t require that I delve into this, it was something I was being persnickety about on my own.
I see that this Khan Academy video treats it as an axiom.
But yes, it’s not one of Euclid’s axioms, but rather a provable proposition.
Commenters have had the luxury of 2300 years to pick through it.
To the extent modern standards of rigor do not allow us to accept Euclid’s proof by “superposition”, you can use something like Hilbert’s proof, in which “side-angle-side” congruence does, in fact, come up as an axiom. (Not “side-side-side” though; that’s still a proposition.) You may as well consider Hilbert’s axiom system a modern foundation for Euclidean geometry.
I explain this to my students that Euclid was writing a textbook and how many K-12 textbooks meet scholarly levels of rigor?
One example of Euclid’s lack of rigor by modern standards is he did not include any axioms related to the concept of betweenness. He just assumed an intuitive understanding of what it means for one point to lie between two others on a line. Hilbert’s axioms took betweenness as one of its explicit axiomatic relations.
Was anyone else in those days treating the subject with more rigor than Euclid?
By the way, Euclid’s proof by “superposition” (and Hilbert’s side-angle-side axiom) may be justified by considering what it takes to have rigid motions in the plane (so you can move triangles on top of each other); see for instance the discussion in Hartshorne’s textbook. It’s not that there is anything wrong in Euclid, only that mathematics has had a few thousand years to become more rigorous
And, really, not only was Euclid more rigorous than his contemporaries, but in some ways, he was more rigorous than anyone for close to two thousand years after him. When Riemann came up with non-Euclidean geometry, the popular headline view was that it meant that Euclid was wrong. But what it really meant was that Euclid was right. Many notable mathematicians in the time between Euclid and Riemann thought that Euclid’s fifth postulate should of course be a theorem, and that it was just a failure on Euclid’s part that he neglected to prove it… except that none of them could actually prove it, either. But Riemann’s work showed that it isn’t a theorem (at least, not unless mathematics is inconsistent), and that Euclid was thus right to classify it as a postulate. And if a different postulate leads to different conclusions, well, that’s what mathematics is all about.
On the other hand, some of Euclid’s other postulates were, in fact, unnecessary. And there were a few hidden assumptions in Euclid’s various propositions, and at least one case where he wasn’t as thorough in a definition as he should have been. He certainly wasn’t perfect. But hey, still pretty good.
As an illustration of why “betweenness” is important, by the way, consider the “proof” that all triangles are isosceles.
I only came across issues such as this “between” thing last year and started a thread about it.
Interesting, and surprising, stuff. But not at all like SSS which is a fairly basic proof. (I would prove it using intersecting circles. Take a line segment the length of one side of the triangles. Draw two circles centered on either end, each the length of the other two sides. Note that there are two intersection points, etc. Or maybe use glide reflections, rotations, etc.)
Around 1900, David Hilbert gave a rigorous set of axioms for Euclidean geometry, which were woefully inadequate. The principle of superposition had no basis in Euclid’s axioms. Probably the biggest gap was the failure to realize that of three points on a line, one is between the other two. The basis of the above-mentioned “proof” that every triangle is isoceles is the failure to recognize that a point in the plane could be inside or outside (or on one of the edges of) a given triangle.
I talk a course in geometry, but that was probably 62 years ago. My recollection was that Hilbert needed twenty some axioms and there was one that something like that if two triangles had all thre sides and two of the angles equal, then the third angles were also equal. From this and the other axioms, theorems like SAS, ASA, and SSS could be deduced. But we didn’t go deeply into the proofs of these facts.
The failure of the all-triangles-are-isoscolese thing isn’t just failing to recognize that the point can be outside of the triangle: You can still construct a “proof” that works the same way for that case. The problem is the order of the points on the other two sides. One is in between two points, the other is outside of the corresponding two points, so you end up with one side being a sum of two numbers, while the other is a difference.
How “first principles” do you want to get?
You have two triangles, presumably one bigger than the other, but congruent in that each corresponding side is the same proportion and consequently the corresponding angles also are congruent (actually, equal)
Simplest way to prove, as if you were Euclid playing with compasses and rulers, is to take sides ABC and sides DEF - extend A to the length of its correspondent side D giving D’, and B to the length of E giving E’. Connect the open ends and you get the same length F=F’
Also note therefore by the parallel lines congruent angles theory, F’ is parallel to C and so angles AC and BC are congruent to D’F’ and E’F’
But the skeptical student says - how do we know F’ is parallel to C? Do we have to prove that a line intersecting parallel lines makes the same angles at both intersections?
Again, using a compass and ruler you can construct parallel lines, so you could construct to show C and F’ are parallel. By contradiction - if two parallel lines are intersected by another line and they don’t make the same angle, then they are not parallel. This is Euclid’s fifth postulate, which has to be taken on faith as there is no contradictory proof.