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?
- 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?
- 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?