[Mathematics] Sum of angles of triangle [Updated]
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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.
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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.
:-O Hello... Basically when you are asking (your self or others) a geometry question you also have to state the nature of the geometry you are referring. As you know, there are a number of geometries. What we have learnt first was Euclidean Geometry that is based on a number of axioms. In this type of geometry, the sum of the angles of a triangle is always 180 degrees. But, for example, in hyperbolic geometry, the sum of the angles of a triangle is less than 180 degrees. So your statement is not always true or false. Depends on context. :) But let's return to Euclidean geometry. You say that no one can draw a line(better said a segment - a line is by definition infinite) of length sqrt(3). Before I go on, I'd like to ask, how do you measure the length? Can you draw a segment of length 1? :). Here you want to mix math with engineering. First of all, in math there are no units. I think they are in physics and fields derived from physics. Anyway, so lets say I can say that given two points we can draw a segment starting from one point and ending to the other (of course, you can ask: how would we know that the line that we draw using those points, contains the points? well, we don't from the engineering point of view. we take this as an axiom. From the math's point of view, a point is something infinite small.). So we have a segment and lets say that it has the length of 1 unit (you name it: inch, centimeter, meter, light-year... :) ). I say I can draw a segment that has the length sqrt(3) like this (using the ruler and compass): first I create a isosceles right triangle having both cathetus of length 1 starting from two points. I consider the segment given by the two starting points as a cathetus and then I draw the rest of the line. Using the compass, I draw a circle of radius 1 and center the second point (B) and get a third point (C) that is antipodal with the first point (A). Now, I draw 2 circles of center A and C with the same radius > 1. These two circles will intersect in two points D1 and D2. If I connect D1 and D2 I will pass though B. Now I will draw a circle of center B and radius 1 that will intersect the segment D1-D2 in 2 points E1 and E2. Well, the segments AE1 and AE2 will have the length sqrt(2). If I repeat the construction using the E1 as B I will obtain 2 other points G1 and G2, and AG1 = AG2 = sqrt(3). I am sure that now you will can draw any segment with length sqrt(n), n > 0. :) Edit: for those out there who
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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
Im agree with you, the angles of any triangle sum 180 degrees, thats math. The real problem is when we want to draw that triangles, our tools and senses has limits. Math is something theoric with theric limits and drawing is something real, with real limits, In theory we can do a lot of things, in the reality we has lot of limits. Think this, you have something with weight of 1 kg in the second floor of a building if you take that thing ando go to the fifth floor the weight will be diferent than 1, because the gravity is not the same, thats another example of the big diference between the teorical world and the real world.
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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 remember seeing this covered in Sphereland[^], a sequel to the classic Flatland, in which one of A. Square's descendants begins to explore Flatland, and discovers many interesting things about this world, including, by way of the very divergence of angles you're referring to, the fact that Flatland is in actuality Sphereland. However, the divergence from 180 degree angles can only be detected between objects that are very far apart - such as stars, as you mentioned. This can only be explained through curvature of space, as others have already mentioned in several subthreads. I hope this helps.
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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
By your own definition perhaps ;) Even if we would use your definition, it still would not be necessarily true. It is the corners that counts here. Whatever happens in other parts of the surface, even if looking only at "the space containing the triangle" does not affect the corner angles. So one can certainly design a surface which has an "average curvature" which is not zero, but which has an "average curvature" of zero if we only consider the three points of the corners. And that would give us a triangle with the angle sum of 180 degrees.
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d@nish wrote:
Which means sum of angles is not 180 degree. Right?
Absolutely NOT! you can accurately measure and construct the angles. The siomplest construction was mentioned by another poster (perhaps you failed to comprehend): 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.
Not only a 30, 60, 90 degree is possible, as it's commercially sold world wide. It's called a 30ยบ square. http://image.shutterstock.com/display_pic_with_logo/87179/87179,1207723301,2/stock-vector-vector-drafting-tools-degree-triangle-degree-triangle-ruler-french-curve-and-a-11307385.jpg[^]
