[Mathematics] Sum of angles of triangle [Updated]
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
One big point the original poster is overlooking... When you deal with angles, etc. a better mode of operation is to use radians. 180 degrees is/(are?) PI/2, not the other way around. SQRT(3) and PI are old friends. They go way back. Also, I think that a lot of this 180 degree calculation goes back to parallel lines which is a postulate that can not be proven. Hyperbolic geometry is the branch of geometry that does not include parallel lines. That is probably what you are after.
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The sum of angles of a triangle is 180 degrees if and only if the edges occur within a surface having zero curvature; that is to say, if and only if the triangle is drawn on a flat surface. If the triangle is drawn on a positively curving surface -- such as the outside of a ball -- the sum of its angles will be greater than 180 degrees. If the triangle is drawn on a negatively curving surface -- such as the surface of a hyperboloid -- the sum of its angles will be less than 180 degrees. This topic is covered in depth in the mathematical field of topology. Cosmological evidence is mounting that the universe as a whole has a negative curvature; that is one of the explanations given for the increasing rate of the universe's expansion. That would mean that a triangle defined by any three points in the universe would, by definition, have angles that added up to less than 180 degrees.
To be strikt, it is not "if and only if". It is definitely "if" but not "only if". Why? Imagine a surface that has both negative and positive curvature, and that you position a triangle in such a way that what you loose in an angle in a negative curvature you gain in an angle in a positive curvature, then it also has a sum of 180 degree. Magnus
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
I could with the same "logic" say that drawing a line where 1 inch represents 1 year, and which has a length equal to the number of years you have lived by now (with all the decimals) is impossible, and thus you do not exist right now. You could counter and say "Well, if 'right now' happens to be when I am exactly 30 years old, then it would be possible". Ok, but what about the sqrt(3) which clearly was somewhere when you were between 1 and 2 years old, then, according to the used logic, it would be impossible. And I could then say that our logic then tells us that you did not exist at that time, and thus you were either not born yet, or you must have already passed away in a tragic early death. This is just one way how the given logic can be used to produce any result we want. Magnus
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
Here are some examples of triangles in Non-Euclidean Geometry... In Spherical Geometry, the sum of the angles of a triangle is always more than 180 degrees. Case in point, a "triangle" formed by the prime-meridian (0 deg longitude), from the equator to the north pole, the north pole to the equator along the 90 deg west longitude line, and the third side is along the equator back to the prime-meridian. All the angles in this triangle add up to 270 degrees! (90 + 90 + 90) Yes, all of the lines are straight lines in this geometry, what are know as geodesics or more commonly, great circle routes. In Hyperbolic Geometry the sum of the angles of a triangle could be more or less than 180 degrees due to the curvature of geodesics in that geometry. (Think of a saddle-shaped "plane"). For a case of less, think of the classical shape of the funnel that you see often representing a black hole in curved space-time. A triangle on that surface would have angles that add up to less than 180 degrees. Hope this helps you understand the strange worlds of Non-Euclidean Geometry. Paul L. Smith GIS Systems Analyst Hal-Tec Software Solutions, Inc.
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I can't believe you guys keep tripping up on Pythagoras while simultaneously writing away as if you knew geometry. I don't know geometry and won't attempt to explain how a triangle might have anything but 180 degrees, but I do at least know that pythagoras' theorem states a^2 + b^2 = c^2 and *not* (a+b)^2 = c^2. Hence with a=1, b=2, c=5^0.5.
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Rob Graham wrote:
On a 2D Plane surface, a 30,60,90 triangle can easily be drawn accurately, but you may not be able to precisely measure the length of the side that is a multiple of the square root of 3.
Which means sum of angles is not 180 degree. Right?
No, the sum of the angles is 180 degrees, you just can't measure accurately the size of the line
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Rob Graham wrote:
construct an equilateral triangle using any convenient side length you wish. Locate the center of one side, then draw a line to the opposite vertex. You now have two perfect 30,60,90 triangles, whose short side and hypotenuse are lengths you know very accurately, but whose long side is some multiple of the square root of 3
more precisely, the long side is some multiple (the convenient length one chose) of the square root of 3/4 (one half root three) I believe the confusion that ripples through this discussion is between accurate drawing and precise measuring. As you note, accurate drawing is simple. Measuring the bisector is tougher, but the limit is in our tools, not our perceptions. We can calculate the length to any arbitrary precision, even though it is an irrational product. Of course the valid precision of the calculated value is limited be the precision associated with original line. An inconvenient proof? :)
there was only ever one purely original computer program, that was the first one...everything since is derivative to some degree - unknown
sandyson wrote:
I believe the confusion that ripples through this discussion is between accurate drawing and precise measuring.
This reminds me of a program that was on UK television awhile ago with Alan Davis and math professor Marcus Du Sautoy. Alan Davis (an actor/comedian) was asked to find the length of a piece of string. http://www.bbc.co.uk/programmes/b00p1fpc[^] The upshot of the programme was: it is impossible to ACCURATELY measure a piece of string; to do so would create a black hole that would engulf the Earth. Now there's food for thought!
No trees were harmed in the posting of this missive; however, a large number of quantum states were changed.
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I can't believe you guys keep tripping up on Pythagoras while simultaneously writing away as if you knew geometry. I don't know geometry and won't attempt to explain how a triangle might have anything but 180 degrees, but I do at least know that pythagoras' theorem states a^2 + b^2 = c^2 and *not* (a+b)^2 = c^2. Hence with a=1, b=2, c=5^0.5.
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
Before anyone else critisizes the pythagoras use in the original post, please note that 1, 2 and 3^0.5 do in fact make a right angled triangle. Why assume the 3^0.5 is the hypotenuse? You can't draw a line of length 3^0.5, because it has no measuring unit, so it isn't a length. But you can easily construct a line of 3^0.5 arbitary units of length by using a reference line of two arbitary units of length, without ever having to worry about measuring the 3^0.5 arbitary units of length line, because you know mathematically it is correct. You don't even need to measure the 1 arbitary unit of length line... If you don't need to know how big the arbitary unit of length is, you don't even need to measure that to know that the line you have constructed is 3^0.5 units of it long. Which means this triangle is certainly possible, and you still don't need to measure the angles to know it'll also have 180 degrees. This construction also creates a perfect pi/2 angle, another irrational number, drawn without ever even needing to know what the value of pi is beyond three and a bit.
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ahmed zahmed wrote:
the square-root of 3 is a non-finite number
It is finite. I think the word you are looking for is "irrational". Another irrational number is 3.14159...
Wikipedia Square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted by The first sixty significant digits of its decimal expansion are: * 1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81038 06280 5580... (sequence A002194 in OEIS) The rounded value of 1.732 is correct to within 0.01% of the actual value. The square root of 3 is an irrational number. It is also known as Theodorus' constant, named after Theodorus of Cyrene. It can be expressed as the continued fraction [1; 1, 2, 1, 2, 1, 2, ...] (sequence A040001 in OEIS). If you can draw a line of 1 unit accurately to .01% of the actual value, I have a job for you :laugh:
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
When you exclude the parallel postulate from Euclid, you get lines that can get closer and closer together, but never intersect. The angeles for traiangles in this space don't add up to 180 degrees. You can read about it from Lobachevsky, or Wikipedia. http://en.wikipedia.org/wiki/Hyperbolic_geometry[^]
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well, I can draw a line of length 1 or of length SQRT(3) equally well as you can not prove I can't draw them. :)
Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles]
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Actually, every line segment I draw is exactly SQRT(3) of some unit in length. :cool:
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
Try this exercise to get a better understanding what this thing about angles not adding up to 180 degrees is about: 1. Take a sheet of paper. Draw an arbitrary triangle on it. Measure the angles and add up. Within a reasonable tolerance, you should get about 180 degrees. 2. Now take this paper and roll it up to form a cylinder. When you look at the triangle, you will notice none of the angles have actually changed. If you want you can glue your cylinder together so you have both hands free to measure them again, but you won't notice any changes. So the angles still sum up to 180 degrees, although the surface this triangle is drawn upon is no longer flat! I'll come to this later. 3. Find a balloon (non-inflated) and some kind of pen that would write on it, such as CD markers. Lay the balloon flat on your desk, and use your pen and a ruler to draw a triangle on it, as big as you can. Measure the angles. This time, due to the uneven surface the sum of angles will probably slightly deviate from, yet still approximate 180 degrees well enough (I'd expect the number you get to be anywhere between 160 and 200). Now inflate it to a nice round ball and tie it up. Measure the angles again. Notice something? All angles suddenly got considerably bigger! Depending on how big your triangle is, your angular sum should now be 50-100 degrees bigger than before, maybe even more. What are the lessens learned? 1. Regardless of how you draw it, a triangle *will* have angles that sum up to approx. 180 degrees when drawn on a flat surface. 2. You may fold, roll up, crumple or otherwise deform a paper you have drawn the triangle on, the angles will not change, nor the sum of angles - this means: A surface does not actually have to be flat for this fundamental theorem of trigonometry to hold! It is still valid on many uneven surfaces! 3. Nevertheless, The theorem stops to be true on many other kinds of surfaces, such as spheres, or balloons. In fact this is true for most non-flat surfaces. The one notable exception is what you've learned in case two: a surface that you can 'wrap up' in a flat sheet of paper (such as a cylinder) is geometrically equivalent to a flat surface. You can not exactly wrap up a sphere in a sheet of paper though, which is why at this point the geometry rules change.
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
ok triangles are out, and pi is not finite so circles are out. i hate shapes.
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Actually, every line segment I draw is exactly SQRT(3) of some unit in length. :cool:
Now we're talking. Don't tell me something can't be done! :)
Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles]
Prolific encyclopedia fixture proof-reader browser patron addict?
We all depend on the beast below.
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To be strikt, it is not "if and only if". It is definitely "if" but not "only if". Why? Imagine a surface that has both negative and positive curvature, and that you position a triangle in such a way that what you loose in an angle in a negative curvature you gain in an angle in a positive curvature, then it also has a sum of 180 degree. Magnus
That would only happen if the average curvature of the space containing the triangle had zero curvature. By definition, an surface with an average curvature of zero has zero curvature, so my original statement remains correct. ;P
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Nope, not at all. The sum of angles in a planar triangle is 180 degrees. Always. Try it out.
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Rob Graham wrote:
On a 2D Plane surface, a 30,60,90 triangle can easily be drawn accurately, but you may not be able to precisely measure the length of the side that is a multiple of the square root of 3.
Which means sum of angles is not 180 degree. Right?
No, it means that the drawing is not truly 30, 60 and 90. Having been a draftsperson in a former life, the advent of CAD vs. hand drawing improved the accuracy. The plastic drafting triangles are approximate, very close, but never 30.00000000000000000, 60.00000000000000000 or 90.00000000000000000 degrees. You could also state, that the sum of the angles in radians is pi. I spend years developing software for AutoCAD that created parametric parts, and I find the concept of using radians for these calculations easier.