Draw any triangle. Pick a side, extend it past the vertices of the triangle. Draw another line parallel to this one, through the triangle's one vertex which is not on the first line. Should be fairly apparent from here.
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Anonymous2007-06-15 4:41 ID:JPJqUb5Z
Draw two horizontal lines. Label three arbitrary points A, B, and C on the upper line, such that B is between A and C, and two points D and E on the lower line. Draw two line segments to form triangle BDE.
Since the horizontal lines are parallel, <BDE = <ABD and <BED = <CBE (where < is supposed to be the angle symbol). Then since <ABD + <DBE + <CBE = 180, therefore <BDE + <DBE + <BED = 180 Q.E.D.
If you make 3 lines who connect to each other from their end, they make 180.
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Anonymous2007-06-15 22:47 ID:tN+YB21N
This is of course only true on a manifold with zero curvature. On manifolds with positive curvature, such as a sphere, the angles will always add up to greater than 180. On a manifold with negative curvature, such as the hyperbolic plane, they add up to less than 180. Look up the Gauss-Bonnet theorem for more details.
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Anonymous2007-06-16 3:02 ID:gPUGXFPd
Actually I think the most scientific of approaching the Triangle is just letting evolution run it's fucking course. It worked for squares, linear equations, and circles, it'll work for us. If you find someidea naturally attractive, aborb information from both your super-ego (Conscience) and id (sex drive and other things) and then let your ego (sense of self) decide how to balance between those two inputs and more importantly your own reality. Then react accordingly, and remember, if you feel bad aboot something, but you want to do it, the three aspects of your personality are clashing and you need to think long and hard and talk to some people you know and develop a set of Mathematical values, that way next time you don't have to think so hard cause you have a set of rules. Whoever succeeds deserves to, whoever doesn't, doesn't. And don't forget how insignificant you really are when it comes to the future of humanity. But also keep in mind that a blizzard is nothing without snowflakes (Nothing more than a cold wind anyway.).
So if you're concerned aboot Triangle, you should be, because that's just how your genetics have programmed your beginnings, and how your surroundings, which are affected by social evolution, have affected your growth. And if the Triangle dies off, whoopdeedoo, we'll just have to try harder with the next sentient species that so inevitably has to arise somewhere in the Universe.
Ow my digression, it hurts my tired mind. Oh yeah, just in case you couldn't catch the moral behind that, it was a scientific way of saying be yourself, not only for the good of yourself, but for the good of your species. And fuck Textbooks with pedagoguery , you make your own shapes constantly, and if someone is attracted to them it means that you are doing the opposite of intellectual inbreeding and would have a wide base of genes for your offspring. So you're promoting brain defects when you read that stuff. Put on normal amounts of scholasticism, or better yet, non-euclidean geometry.
And yes I am a triangle virgin, because I would not want my son telling a highschool girl about things when she is not nearly ready enough for these kind of emotions while she's still getting used to her intellect. So I am avoiding hypocrasy by waiting until I can view myself as my own father and approve. Whether or not I could have decided about triangles by now had I been trying to like so many other id-driven dogma monkeys my age, is a total mystery. But not really, because I am who I am and it's not like that's going to change, the purpose of the last few points was to say that I'm not the kind of guy that says: "I could've had triangles if I wanted to PPPBbbbPPTTbt"
I haven't slept in 3 days, and I'm going to. Right. Now. Goodnight.
>>1
The sum of the outer angles of an arbitrary convex n-sided polygon have to add up to 360. The inner and outer angle of a vertex together sum to 180, so the sum of the inner and outer angles is simply 180*n. Hence the sum of the inner angles is 180*n-360.
For a triangle, n=3, so 180*3-360 = 180.
That's also why for an arbitrary convex quadrilateral (such as a rectangle or trapezoid), the inner angles add up to 360.
Nice. But layman might not understand why the outer angles of a "convex n-sided polygon have to add up to 360" so I'll just point out that it can be inscribed in a circle.
Take an equilateral triangle ... you know, all 3 sides are equal, hence all 3 interior angles are equal. Pull one of those corners out to infinity. That angle at infinity is 0. The stretched sides are now parallel, so their base angles (i.e. the ones by the side that was not stretched) are 90 degrees. 2x90=180.
Visually, the layman should see that as long as the triangle is flat, you can stretch those sides all you want, but the sum of the interior angles is conserved. Hence, that 90+90+0 sum can become any other triplet of values, as long as they add up to the same result: 180 degrees.
You could also use the flat triangle to make the same point. Take the equilateral triangle, then compress it, making one side infinitely long. The opposing angle becomes flat (hence a 180-deg angle) and 2 adjacent angles go to 0. Hence, 0+0+180=180.
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Anonymous2007-06-17 13:27 ID:8g8x3uOw
>>14
No it can't (e.g. parallelogram), and that's not even a necessary condition; the outer angles of *any* convex polygon, cyclic or not, add up to 360. (actually, I think it works for any simple polygon provided you consider the outer angles of concave vertices as negative).
Take one vertex, and extend one of the lines outside. The angle between the extended line and the line on the quadrilateral is the outer angle of the vertex. But see, that line on the quadrilateral is the same one you extend to get the next outer angle. You go all the way around when you get back to your original vertex, so 360.
Horrible explanation, I know, but whatever, I'm tired.
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Anonymous2007-06-17 13:57 ID:Z5r+Xz5w
Triangles add up to 180 degrees on a flat plane in Euclidean style geometry only as a matter of simple arithmetic when you enclose a space on that flat plane within 3 straight lines whose ends connect to form angles and you add those angles. To make the angles add up to some other number, warp the plane so that it is no longer flat or keep the plane flat and warp the triangle sides.
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Anonymous2009-03-18 2:52
I'm feeling really keen, for some of that good ol' green
Marijuana MUST be legalized.
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Anonymous2012-11-15 20:25
What about spherical triangles?
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Anonymous2012-11-17 10:34
A triangle is half a rectangle
Rectangle = 360 so triangle = 180