$\begingroup$

TL;DR Triangles helped me understand both unit circle and trig functions. Super cool and super useful. Didn't really use much of my other geometry. Don't drop triangles. If you do, you should have a good reason and better replacement.

I am not a mathematics educator. But I did take junior high geometry that covered all those topics. Here's a student's perspective.

I don't see a topic on that list I regret studying. I always thought studying triangles in-depth made it easier to appreciate the role of triangles in defining the trig functions using the unit circle. I took Trigonometry freshman year of high school. If I had just learned SOHCAHTOA alone, I would have never appreciated the important the properties of right triangles versus nonright triangles. For me, my first look at trig was seeing how geometry could give me a new powerful mathematical tool (ie sin cos tan) through the geometry I has learned. Thus, understanding many properties of triangles gave me a better appreciation for how trig functions are defined and why right triangles are useful for that. I thought trigonometry was beautiful in part because I was so familiar with triangles.

That said, I feel my future math courses never used much of the geometry I learned in that course. I have never used SAS or SSS ever since then. In fact, triangles were one of the few things I actually used after that course. Only many years later when I began studying differential geometry and non-Euclidean geometry did I really feel a curiosity to revisit the principles I learned in my geometry class. And I personally always was disappointed by this. Now that I reflect on it, most courses I took in high school I have used. Geometry would be an exception in that much of it I haven't used.

So perhaps if curriculums made more use of non-triangle concepts, it would be worthwhile to study stuff besides triangles. But, especially if you do physics or anything involving Fourier methods, you want to have a good grasp of trig and I feel triangles are a key piece of that. In other words, I wouldn't change how triangles are taught unless I see topics I know will be more useful that should replace them.

I actually would have liked to study stuff besides triangles more in-depth. So I am not necessarily pro-triangle. But I am always trying to invest my time in things that are useful and triangles clearly were the most valuable part of my geometry course.

EDIT 1:

All of the above was, so to speak, in defense of triangles. But suppose there are better things to teach. Like what? If teachers want to teach less about triangles, I would be interested to hear a proposal for what should be taught in place of them. This proposal should keep in mind that secondary school isn't designed to train mathematicians but to provide children and young adults mathematical tools that will be useful for their professions. This ranges from finance to physics and beyond, so we aren't looking for merely the most interesting math but the most useful. So such a proposal should (1) specify what use the math would serve and (2) why this is more useful than studying triangles.

EDIT 2:

I also think you could simply break up the triangles into separate units instead of bundling them together. This would allow you to teach other topics in between. For instance, split off together the trigonometric properties of triangles as well as the laws of sines and cosines. Make those a separate unit and do that unit later in the year. It should be workable I would imagine without having to drastically refresh the students on everything.