# Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

The assumed age of the students is 10-15 years old.

What is the danger in saying that two triangles are "the same" or "equal" instead of saying that they are congruent? It seems to me like the term "congruent" is an unnecessarily rigorous term to introduce to children at that age when we already have vocabulary that seems to work just fine.

Intuitively, at least for me, what the concept of "congruency" captures, that "sameness" might not, is that the former includes the notion of triangles being mirror images of each other. However, for triangles this distinction does not seem to matter fully, as you can still move (in 3-space) any two congruent triangles so that they overlap.

So why do we really introduce the notion that two triangles are "congruent" instead of just saying that they are "the same" or "equal"?

• For what it's worth, I don't believe I encountered the word "congruent" in a math class (I'm not counting seeing it in independent reading or otherwise outside a formal classroom setting) until my 10th grade geometry class, where all of the students were 15-16 years old. Jun 4 at 5:34
• I feel like there's an outside chance that this question reveals a weakness in understanding what "equal" really means (which is a very common issue). That means the two compared objects are absolutely identical in all respects. E.g., for geometry, it would imply same position in space. Jun 4 at 13:49
• It is best to say "In a square all sides are congruent" and not to say "In a square all sides are equal". Even though all points are congruent, It is REALLY not good to say "All points are equal." Jun 4 at 15:27
• An example of a profound use of the power of congruence over "sameness" is Pappus's proof of the isosceles triangle theorem en.wikipedia.org/wiki/Pons_asinorum#Pappus. A triangle and its mirror image aren't generally, in common sense, the same, but they are congruent. Jun 4 at 18:10
• In my native Bulgarian (as well as in a few other languages) the coungruent triangles are called "same". No problem as long as we got the definition of this "same". Jun 6 at 16:42

Colloquially, there's a lot of conceptual overlap between all of these terms, but "sameness" is not a well-defined mathematical property. Congruent shapes need not be "the same" or "equal" in all respects - they can be rotated differently, or be in different positions, or be different colors, or have different names, or differ in any other characteristic that's not size and shape.

Congruency defines exactly which properties need to be "the same" for two shapes to be considered as such. "Sameness" is an ill-defined concept that doesn't describe which properties should or should not be the same in order to qualify, and "equality" somewhat implies that all properties must be the same, which also isn't quite true of congruency. Congruency is a precise term that indicates equality of side lengths and angles and their arrangement, but nothing more or less. If you want to describe congruency as "sameness" you'd need to elaborate on exactly what characteristics need to be "the same" in order to qualify, in which case you're just back to the definition of congruency.

One could reasonably argue that two congruent triangles drawn in different colors are not "the same". You could also argue that two congruent triangles drawn on different parts of the page and rotated differently are not "equal" to one another (as "equality" of shapes isn't defined in the first place). Congruency is the right term to describe exactly what properties you're considering. "Sameness" or "equality" are reasonable terms to introduce the general notion of congruency in a non-rigorous way, but they are too imprecise and open to interpretation in a way that "congruency" is not.

• +1 for a very good answer; I'd go on from your third paragraph that more than a reasonable argument, but there is a whole school of thought (computer science) wherein if you have two triangle objects, they can never be "the same/equal", regardless of colors or location; even if they are in all ways identical two objects are not the same, because they occupy different memory locations. I would note congruency is exactly how a TA made this concept stick for me. Jun 4 at 18:32
• @sharur this is a good point. For those unfamiliar, in CS, we talk about identity and equality. Two things are identical if they are absolutely the same thing. For complex entities, we call them equal if they fulfill some criteria for matching... exactly what these criteria are depend on the implementation of the code. In mathematics, no two shapes are ever really "identical". There's always something that will differentiate them, otherwise you couldn't really point to two of them. Congruency is mathematic's "implementation" of equality. Jun 4 at 19:02
• Excellent answer! This also explains why there is a need to say things like two groups are "isomorphic" rather than "equal", or that two topological spaces are "homeomorphic", etc. Jun 5 at 12:42
• @GitGud But that's not how English works. Saying that two things are the same is not saying that they are the same thing. Jun 6 at 1:46
• @Araucaria-Nothereanymore. Actually, that's how English works in some technical fields, specifically in computer science and mathematics. There could be something to your point had sharur said "two variables" or something akin to that. But they didn't, they were very specific about having two different objects. This isn't colloquial English, this is technical English. Jun 6 at 9:28

The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually "the same" or "equal". Usually when we say two things are "the same", we mean that they are not just indistinguishable, but that they are literally the same exact thing. "Equality" means two numbers are the same, or two sets have all the same elements as each other. So "sameness/equality" for geometric figures means that every point of one figure is also a point of the other figure. But this definition is obviously too restrictive if we instead want to talk about two figures that have the same size and shape, but might be different considered as sets of points, as is typically the case in geometry.

You can imagine the confusion for a beginner student in geometry if they were presented with constantly moving goalposts for when two figures are "the same", where sometimes we literally meant they have all the same points, and sometimes we only meant that their measurements are equal in every respect (or equivalently, that a sequence of rigid motions exists to put one atop the other). Or on the teacher end, having to deal with a smart-aleck student going, "Those triangles are obviously different triangles, one's over here and the other's over there; why are you saying they're the same?" So it's convenient to have a shorthand word for "same size and shape, but not necessarily same points," when that's actually what we mean.

• Interesting point about different interpretations of "the same", as in "I'm wearing the same shirt I wore yesterday" (referring to one object at two points in time), versus "you and I are wearing the same shirt" (referring to two objects with identical properties). I wouldn't say the former interpretation is necessarily more common, but the dual interpretations do make it an imprecise term - it indicates similarity, but not in what characteristics (object identity, ownership, shape, color, size, etc). Jun 4 at 16:09
• There is also the human/civil rights angle for equality. When it comes to enrolling in math classes, girls are equal to boys (or ought to be, if there are still schools where they are not). This doesn't mean that they have the same points or measurements. Jun 5 at 14:22
• Saying that two things are the same does not mean they are the same thing. Jun 6 at 1:49
• @araucaria Yeah, but as I pointed out above you would sound like an idiot telling smart-aleck HS students, “‘the same’ doesn’t actually mean ‘the same’,” and you’d almost certainly have to spend a lot of time clarifying, qualifying, and (to some extent) walking back that initial statement. We have the word “congruent” in order that we never have to say any version of such a confusing, and apparently self-contradictory, sentence to beginners. Jun 6 at 21:33
• "The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually 'the same' or 'equal'." I don't think that's a smart-aleck answer, I think it's just the answer. Jul 25 at 6:41

Historically (and by historically I mean "in Euclid's Elements") the word "equal", when applied to geometric figures, meant "equal in magnitude". So for example:

• Euclid refers to two segments as equal if they have the same length
• Two triangles are equal if they have the same area
• Two solids are equal if they have the same volume

For example, Elements Book I, Proposition 35 says:

Parallelograms which are on the same base and between the same parallels are equal to one another.

(See figure below, from Fitzpatrick's English translation of the Elements, based on the Greek text of Heiberg.)

Notice that "equal", in this usage, does not mean "congruent"; nor does it mean "the same, identical". With this idea as background, the word "congruent" plays an important role: it allows us to say that two figures are not only equal in area but are precisely the same shape, i.e. can be superimposed onto one another.

Language, of course, is constantly evolving. At some point people stopped using the word "equal" to mean "equal in area", and began using it to mean "the same mathematical object". I am not sure exactly when this switch happened, but I suspect it was in the latter part of the 19th century, as set theoretical ideas began permeating all aspects of mathematics. The greatest sea change in Geometry during this time was Felix Klein's Erlangen Programme, which sought to reframe Euclidean Geometry as the study of properties that are invariant under isometries of the plane. In this context it is important to distinguish between the statements "$$\triangle ABC = \triangle XYZ$$" (which means that the two triangles are the same mathematical object) and "$$\triangle ABC \cong \triangle XYZ$$" (which means that there exists an isometry mapping one triangle onto the other).

The effect of this linguistic switch was that the relative valences of the words "equal" and "congruent" became reversed: whereas in Euclid's work the word "equal" is a weaker relation than the word "congruent" (in that congruent figures are always equal, but the converse is not true), nowadays the word "equal" describes a relationship stronger than congruence (in that "equal figures" are (trivially) congruent, but the converse is false.)

So there is one reason why it is important to have different words: to distinguish between different notions of "same" ("same area" vs. "same figure" vs. "same size and shape").

I would go even further, and say that a lot of the vocabulary we use in Geometry is introduced to disentangle notions that, to a naive student, seem equivalent.

• For example: to many students, a statement like "rectangle $$ABCD$$ is bigger than rectangle $$PQRS$$" seems perfectly sensible. That is because they have an unexamined notion of "relative size". But it is possible for one rectangle to have a larger diameter than another, and simultaneously a smaller area, and equal perimeter! (Which is larger: a $$6 \times 8$$ rectangle, or a $$7 \times 7$$ rectangle?) So we introduce vocabulary (area, perimeter, diameter) in order to unpack these distinct notions of "size".
• A second, more sophisticated example: if you ask students to find a point in the interior of a triangle that is equidistant from all three vertices, they will say "in the center". If you ask students to find a point in the interior of a triangle that is equidistant from all three sides, they will also say "in the center". It seems both obvious and unproblematic to them that "center of a triangle" means something -- and that it means one thing. But in fact there are multiple notions of "center", all of which are worth naming, and therefore we need different names (circumcenter, incenter, orthocenter, centroid) to distinguish them.
• @user615 quite right, I have amended my first sentence to incorporate this correction. Jun 9 at 1:02
• Regarding different notions of size, you might be interested in Matthew Wright's video on Hadwiger integrals. He starts by asking "What is the size of a box?"
– J W
Jun 9 at 4:53
• This hits the nail on the head as to the role of language. Some other answers and comments seem to show less awareness of the fact that the correspondence between words and mathematics is in part arbitrary and a matter of linguistic convention. One should also keep in mind that conceptualizing a triangle as a certain point-set in R^2 is not the only way of forming a foundation for mathematics. In smooth infinitesimal analysis, a geometrical curve or figure is not a point-set.
– user507
Jul 22 at 17:37

The problem is that there are different kinds of 'samenesses' (equivalence relations). One of these is congruency, but another is similarity. If you don't teach your students the word congruent, and use the word same instead, what word are you going to use when introducing similarity, since similar triangles can also be thought of as being the same, but in a different way than congruent triangles. It is important for the students to understand that the concept of being the same is captured by multiple different notions in mathematics.

I agree with the point in the other answers, that the difference is in the approach, how rigorous we would like to be.

However, I would like to also mention that this rigour in discussing elementary geometry is not extended to some other parts of teaching mathematics in high school. I do not think that too many high school teachers discuss vectors in a rigorous way (for example as equivalence classes of directed line segments). I think it is pretty common to refer to directed line segments of opposite sides of a parallelogram as equal vectors or same vectors.

I guess, the difference is the goal. Elementary geometry is a good topic where axiomatic, rigorous approach can be introduced in a way that is accessible to high school students, so it is natural to insist on formal language.

I can tell about how it went in Russia, presumably, that might have gone the same way in other countries as well.

Traditionally, congruent shapes were called "equal". Then, in the 1960's, it was felt that some set-theoretic notions should be introduced into the school geometry, a charge led by leading mathematicians such as Kolmogorov. Since there's a notion of equality for sets, and "a triangle is a set of points, but two triangles being equal does not imply they are equal as sets" sounds contradictory, a notion of congruence was introduced.

The whole project was, however, a complete disaster, in part because of genuine failures in the implementation, but mostly because of lack of qualification or desire in the teacher's community to switch. There was a huge backlash, with the word "congruent" one of the targets of mockery. So after some time, it was rolled back, and textbooks reverted to "equal triangles".

But the problem is, if you want to give any definitions at all, then a modern definition of e. g. a circle would likely contain words like "a set of points". And it is hard to justify "definitions" like "line has only length but no width" when rigorous definitions are nearly as easy to state. So, some definitions, and some set-theoretic language, and hence also congruent triangles crept into some textbooks. It helped that new generations of teachers were raised for which the notion of a set is not as alien as for the old guard.

So, as it happened, in the middle school we studied equal triangles, then, in the high school, we were told that it is more correct to call them congruent (without much explanation), and later on we were introduced to some set-theoretic language that did explain the reason. But I understand that some textbook authors prefer to avoid that switch and introduce the proper terminology from the beginning.

This is much less rigorous, but I taught my students that numbers or measures can be equal, but objects--like segments, polygons, and angles--cannot be equal, they must be congruent. It was a distinction I gave particularly to help with proofs, where they often need to say two angle measures are equal (and use the equal sign) before saying the angles themselves are congruent (with the congruent symbol).

The same triangles can always be put on top of one another, while congruent triangles can't, generally. If we confine ourselves to the 2d plane, that is. Putting on top requires the third dimension though, so let's state that they can be put on top of one another without rotating them around an axis in the plane.

If a triangle has the same three angles as another angle they are said to be congruent. The lengths of their sides need not be equal though. So two congruent triangles need not be the same.

Even if the lengths of the sides are the same, the triangles need not be the same. They can be mirror images of one another.

So congruence of triangles is a broader vision of triangles as the same triangles. Congruent triangles can have different side length (though the ratio of corresponding sides is the same), they can be mirror images, or they can be rotated wrt to each other.

• I am not the DV, but kindly saying, this answer is wrong. 3 equal angles results in “similar” triangles, not congruent. And two congruent triangles may very well be reflected over a line, i.e. mirror images. Jun 6 at 13:22
• @JTP-ApologisetoMonica All triangles with the same angles are congruent (they can be similar or the same). As I wrote in the answer, they can be mirror images of one another (as you wrote too). To are right though about the fact that they have to have equal lengths. Congruent triangles have the same lengths and the same angles (contrary to what I wrote). So the same triangles can be put on top of one another, while congruent triangles don't, generally (they can be mirror images). This mirror property is the only difference between congruent triangles and the same triangles. Jun 6 at 13:36
• Sorry, just no. All congruent triangles are similar, but not vice versa. Similar means same angles (for triangle, not other polygons) but need same length for congruent. Jun 6 at 13:53
• @JTP-ApologisetoMonica You are right! I thought congruent meant that only the angles have to be equal. But also the lengths have to be equal... Jun 6 at 13:55
• I was bracing myself for you to say that in a non-US country the words are used differently. Thanks for this update! Jun 6 at 14:04