Why do standard geometry textbooks not start with trigonometry?

Question

Throughout my geometry course, I was given many theorems and postulates, which I was were expected to memorize and apply. At the time, I sorta went along with it, but I couldn’t help but wonder where these came from. Probably 90% of these questions were answered when I took trigonometry.

Take, for example, proving two triangles congruent. As we all know, one can prove them congruent by side-side-side, side-angle-side, angle-side-angle, angle-angle-side, and hypotenuse-leg. My teacher, I distinctly remember, made sure to emphasize to “stay away from the bad word, forward and backward” (angle-side-side).

But wait! I wondered at the time. Why doesn’t angle-side-side work, but hypotenuse-leg does? Isn’t hypotenuse-leg just a special case of angle-side-side, where the angle is a right angle? To this day I cannot come up with a rigorous proof that justifies these that doesn’t rely on the fact that $\sin\theta=\sin(\pi-\theta)$, meaning that, unless $\theta=\frac\pi2$, there are two possible side lengths corresponding to the same angle, and thus two triangles that contain the same angle-side-side but differing third sides.

In general, I think, explaining the basic trigonometry functions and the unit circle as the very first thing in the course would make pretty much all of geometry make so much more sense. Granted, even the unit circle requires certain geometric knowledge (ex. definition of a circle, that all radii are congruent, etc.), but not too much that would seem to preclude some form of this working?

1. Is what I’ve described a typical geometry course worldwide, or at least US-wide, or did I happen to end up with a poor geometry course?
2. If this is the normal way of teaching geometry, why? Why is the course focused more on memorizing theorems rather than understanding where they come from — something which, in my experience, seems to be nigh-impossible without the tools learned after Geometry?

Answer 1

If you really want to understand why the curriculum is structured the way it is, and how it got that way, you might want to read a brief history of the discourse around geometry education for the past 150 years or so; I recommend Chapter 1 of The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective, by Herbst, Halverscheid, Fujita, and me.

However, I think the question actually relies on a (pardon the pun) circular argument: Certainly one can introduce $\sin, \cos, \tan$ etc as "circular function" defined via the coordinates of points on a unit circle. But in order to make use of those functions in triangles, you have to know how to adapt them to the case when the hypotenuse is not $1$. How do you know that the circular functions are scale-invariant?

The answer, of course, is that you rely on similarity properties: specifically, if two angles of one triangle are congruent to two angles of a second triangle, then the ratios of corresponding sides are equal ("Angle-Angle Similarity"). To solve any complicated problems involving triangles, you also need to know that it works in the other order: if the legs of one right triangle are in the same ratio as the legs of a second right triangle, then the corresponding angles are congruent (a special case of "Side-Angle-Side Similarity"). So if you really want to start with trigonometry, you have to preface it with a unit on similarity.

But similarity properties (and their proofs) depend in an essential way on the properties of parallels, so you can't really cover similarity without first covering parallels. And at that point you have basically reproduced about 2/3 of the first few chapters of a traditional geometry course anyway, so why not do the whole thing?

Answer 2

I cannot speak to curricula outside of the US, and I don't really buy your argument that geometry courses are "focused more on memorizing theorems rather than understanding where they come from." It is a poor geometry instructor indeed who admonishes students "Don't be an ASS" before first taking the time to discuss the failure of the angle-side-side congruence relation. Because I don't find much in your question that I can directly respond to, let me start by questioning the framing of your question, and instead answer the question of why geometry is taught the way that it is (which is, after all, the essence of the title of your question).

My impression is that geometry is usually taught (in the modern world, at least) because it is a relatively concrete and hands-on way of getting students to engage in mathematically rigorous argumentation. Very few people actually care if two triangles are congruent or not, or how to construct a regular hexagon. However, the process of proving statements about triangles and hexagons introduces students to the idea of mathematical logic.

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In the example you give, I would hope that the pedagogy follows something like the following outline (modulo some permutation of ideas, depending on the instructor, the author, or where the students lead you):

1. We observe that a triangle has three sides and three angles. To a mathematician, it is (I think) natural to ask how many of these data are actually necessary to uniquely specify a triangle, up to congruence.

2. So, we get out our compass and straight-edge, and start doodling. Experiment a little. If we are clever, we should quickly discover that if two angles are specified, then the third angle is completely determined. This gives us the Angle Sum Theorem, which we can prove satisfactorily in the Euclidean setting.

3. However, we might note that three (or really, two) angles are insufficient to uniquely specify a triangle. Two triangles can have the same set of angle measures, but be of radically different sizes. However, such triangles will be similar (i.e. there is a constant ratio between the lengths of corresponding sides). We can prove this result, and, perhaps, call it the AAA Similarity Theorem (for angle-angle-angle).

4. At this point, we know that if we have determined two angles, we need at least one side length in order to uniquely determine the triangle. The natural question here should be "is this sufficient?" That is, if I know the measures of two angles and the length of one side, can I determine everything else about the triangle? The answer is yes, giving us an AAS Congruence Theorem (or, in your question SAA, for side-angle-angle).

5. Now we might want to know if we can lose even more information and still make progress. For example, is there an AS theorem? That is, if we know an angle and a side, is that enough to determine a triangle. Very quickly, we should see that the answer is "No"–counterexamples are fairly easy to construct. Fair enough. If we only know one angle, knowing the length of only one side is insufficient.

6. Here the real fun starts: if we are clever (or have good guidance), we should start to realize that if we know an angle and two sides, then the arrangement of those objects matters: if the angle is "between" the two sides, then a triangle is uniquely determined, a result which we can prove as the SAS Congruence Theorem. On the other hand, if the angle is not "between" the two sides, we get into trouble.

7. The mathematically inclined might ask to classify the ways in which we can get into trouble. After some investigation, we should determine that there are essentially three things which can go wrong (assume that $AB$ is a side with a specified angle at $A$; we wish to construct the side $BC$ of specified length):

   • We construct a circle of the specified radius centered at $B$, but the radius is too small, and so we cannot construct any triangle with the specified angle and sides.
   • We construct a circle of the specified radius centered at $B$, but the circle intersects the the opposite side of the angle at $A$ at two different points, leading to two possible triangles.
   • We construct a circle of the specified radius centered at $B$, and the circle intersects the opposite side of the angle at $A$ at exactly one point, giving exactly one triangle. There are actually a couple of ways that this can happen: either the specified length is longer than $AB$ (in which case the circle will meet the line at only one point on the "correct" side of the the angle), or the circle will meet the line at a point of tangency (in which case, the side $BC$ will be perpendicular to the side $AC$).

8. Because there is ambiguity in the "ASS" case, we don't get a nice theorem, and we demand more information in such a case. However, do note that the "ASS" case is not always indeterminate: for example, we get the Hypotenuse-Leg Congruence Theorem as a special case.

This topic could be further expanded (it might be nice to carefully consider exactly when ASS fails and how and what extra data may be used to patch things up), but I've already hit the main points alluded to in the question, so I'll stop here.

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Having laid out an outline for a possible series of lectures or activities for students, let me make the following points:

• There are (by my count) five theorems here which require proof. Real mathematics is about rigorously proving statements, so there are a number of good exercises here for students to work on. As an added bonus, each of the theorems to be proved should be more-or-less intuitively obviously true, and good constructions will help the students to see how to make the arguments. In higher mathematics, it is sometimes hard to visualize an argument, and it is often difficult to tell a priori whether or not a result is true (there are a lot of "obvious" algebraic statements that are a pain to prove in an abstract ring, and a lot of wicked estimates in analysis which are hard to visualize or otherwise understand intuitively), hence Euclidean geometry is a good sandbox for learning to pose and prove theorems.

• There are a number of cases where things don't work. This can be used to emphasize the power of a counter-example. Once we show that there is an example in which ASS fails to produce a unique triangle, then we know that no ASS congruence theorem can possibly exist. Counter-examples are amazing! Indeed, two of my favorite books are slim Dover volumes full of counter-examples in analysis and topology. However, these require a lot more technical knowledge than a geometric construction. Again, Euclidean geometry is a good sandbox for learning about counter-examples.

• Finally, to reiterate, no one really cares about most of the results in a high school geometry class. By suggesting that we skip straight to trigonometry, which is its own nightmare of abstract notation and endless confusion for students, you are suggesting that the goal of such a class is to learn computational techniques for determining angles and sides (or whatever). This completely misses the point, which is to explore a topic in a mathematically rigorous manner. The results are not at all important (for the most part); it is the process by which those results are obtained that is important.

Answer 3

Here is a late but brief answer to the question:

If this is the normal way of teaching geometry, why? Why is the course focused more on memorizing theorems rather than understanding where they come from — something which, in my experience, seems to be nigh-impossible without the tools learned after Geometry?

This is not normal. What the OP describes (lots of memorizing and no justification) is, I'd argue, the exact antithesis of what the standard geometry class is for in the curriculum -- to give students a taste for rigorous mathematics, where principles are built up from axioms, without taking everything on faith.

Basic geometry does not require tools from trigonometry for its development, and historically did not use them. To address one example, here is a proof of the hypotenuse-leg theorem using only elementary geometry.

The prospect of angle-side-side congruency can be disproved with a single counterexample, of course.

Answer 4

My mental picture of why angle-side-side doesn't work as a congruence condition doesn't involve trigonometry and seems to me much simpler than trigonometry: Imagine that you're given a line segment $AB$ to serve as one of the two specified sides of the triangle and you're given the angle at $A$, so you can draw a line $l$ through $A$ in the desired direction. The third vertex $C$ of the desired triangle should be on $l$ and should be at a specified distance from $B$. That gives you a circle $m$ of possible locations for $C$, centered at $B$, and $C$ must be at an intersection of $l$ and $m$. In general, a circle will intersect a line twice; for the two choices to produce congruent triangles, you need that they're equidistant from $A$. In other words, the chord of $m$ joining the two intersection points should be bisected at $A$ by the radius of $m$ that passes through $A$. But a chord of a circle is bisected by a radius if and only if it's perpendicular to that radius, and that's why the angle-side-side condition works iff the angle is a right angle. (Strictly speaking, one should pay attention to the case where $l$ meets $m$ only once, i.e., where $l$ is tangent to $m$, but there again you have a right angle.)

So this example doesn't need (or, in my opinion, even benefit from) trigonometry. Offhand, I can't think of other examples either that would justify putting trigonometry earlier in the curriculum. Trigonometry is a useful tool, but the essential ideas of geometry don't involve it.

Answer 5

Anecdotal evidence from Germany: congruence is taught way before trigonometry (~7th grade vs ~9th grade for trig functions)

Why is the course focused more on memorizing theorems rather than understanding where they come from.

The fact that two triangles are congruent if their sides are pairwise of equal length has nothing to do with trigonometry in the first place; one can give a constructive proof (that also shows why the triangle inequality must hold) without trigonometry.

To see the same fact using trigonometry one would have to apply the law of cosines twice which in my eyes is much harder. This law will also most likely be memorized without "understanding where it comes from" because the proof is again much harder than the constructive ones for congruence using compass, straightedge, and protractor (In Germany, we call this a Geodreieck).

Answer 6

I can relate to your observations! Most textbooks start the same ... Points, Lines, Quadrilaterals, Polygons, and finally ... the triangle. However, what if we started with the triangle and the circle and noted all the other concepts taught first and apply it to the triangle in particular... particularly noting that triangles are and can be inscribed in so may other shapes invisible to most...use triangles to express squares, rectangles, parallelograms, etc.

I am definitely going to try this methodology with some of my high school students next year!

Maybe it is time to think inside and outside the triangle!