voidPtr wrote:One thng I don't ever remember being proved in grade-school maths is that the angles in every triangle add up to 180 degrees.

Actually, I remember a lot more of what wasn't proved than what actually was proved, some of it for good reason, some not.

Actually, I rememebr this one being shown specifically to me when I was in school. School grade "proof" follows:

Imagine walking round the edges of a triangle. Suppose you start at a vertex of the triangle, and walk along one of the edges until you reach the next vertex. You now rotate, through the smallest angle possible (i.e. the exterior angle), so that you are facing in the direction of the next edge. Let's call the amount you rotate to do this, in degrees, [imath]r_1[/imath]. You carry on your journey, repeating this process at the next vertex (rotation [imath]r_2[/imath]), and finally at the third vertex (rotation [imath]r_3[/imath]) so that you've gone all the way round and are now facing the same direction as whe you began.

As you've turned in a full circle in the course of talking your walk, we know

[math]r_1 + r_2 + r_3 = 360[/math].

Now consider the interior angles at each vertex of the triangle, calling them [imath]a_1, a_2, a_3[/imath]. As each one, when added to the exteroir angles (the ones you rotated through) must add up to 180 degrees (as together they must make a straight line), we now know

[math]a_1 = 180 - r_1[/math]

[math]a_2 = 180 - r_2[/math]

[math]a_3 = 180 - r_3[/math]

And so we can get the sum of the interior angles like so

[math]a_1 + a_2 + a_3 = 180 - r_1 +180 - r_2 + 180 - r_3 = 540 - (r_1 +r_2 + r_3) = 540 - 360 = 180[/math]

so the interior angles of any triangle must add up to 180.

I've not thought about that for years. With hindsight I'm impressed at my maths teacher for going through it with us, as I'm doubtful it would have been on the syllabus.