[Mathematics] Sum of angles of triangle [Updated]
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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[^]
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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.
In 2D any triangle's angles sum is going to be 180 no matter what. Put the same theory to 3D and it fails. Imagine drawing triangle on a ball. The lines are going to be curved and in that case the sum of angles of that triangle is not going to be 180. page 128 in the book "Mathematics, The Science of Patterns" by Keith Devlin, a Scientific American Library book. There was the picture of a pseudosphere and on that the angle sum of a triangle is less than 180.
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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.
Pretty interesting to find out. It is always true that sum of angles is always 180 degree if and only if the triangle is lied on a flat surface(plane). Likewise, however, Hyperbolic & Spherical trigonometry don't show this kind of attribute. Hyperbolic triangle can have sum of angles less than 180 degree, whereas Spherical triangle can have more than 180 degree. These would be another case of triangle whose the plane is not flat.
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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.
Actually, the vertices are irrelevant. What are important are the ANGLES, as that is what we are measuring. If the angles of a triangle add up to 180 degrees, then space containing the polygon has zero curvature, by the operative definitions of zero-curving space. Space elsewhere, away from the polygon, might have a different curvature, but that is also irrelevant.
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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.
This is a bit silly, really. You are equating your ability to draw a triangle with the validity of a Euclidean geometric rule! In Euclidean geometry the sum of the angles of a triangle MUST add up to 180 degrees. However, you can have other geometries where this is not true. For example, Lobachevsky created a geometry in which Euclid's 5th postulate (that for any given line and a point not on that line, there is one parallel line through the point not intersecting the line) was false, that is, there is more than one line that can be extended through any given point parallel to another line of which that point is not part. One of the consequences of this is that the sum of the angles of a triangle must be LESS than 180 degrees. This geometry was an idle curiosity for over 100 years until it was found to be useful in relativistic physics.
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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.
Hello, you can draw this line, and you stated how: - Draw two perpendicular lines. - Take your compass and open it some width; it defines a unit. - With your compass measure, from the intersection of the two lines, measure one unit right and two units up. - Link the two resultant points with a line. This line measures SQRT(3). So you can effectively draw this line. Sum of angles of a triangle is 180 degree, yes, in an Euclidean geometry (plane geometry). But in other geometries this sum can be bigger, take this example: - Take the sphere as the surface of this geometry. - A "straight line" is the line that joins two points with the minimum length, in this case is a circle of maximum length. Over the sphere there are no paralel lines. - Consider the triangle defined by: the North Pole, the Ecuador at meridian 0, and the Ecuador at meridian 90. In this triangle all three angles measure 90 degrees. I cannot recommend you any bibliography, I'm not an expert on Geometry. But some articles about non-Euclidean Geometry in Wikipedia could be a good start in this theme. Regards, Francesc
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Actually, the vertices are irrelevant. What are important are the ANGLES, as that is what we are measuring. If the angles of a triangle add up to 180 degrees, then space containing the polygon has zero curvature, by the operative definitions of zero-curving space. Space elsewhere, away from the polygon, might have a different curvature, but that is also irrelevant.
> If the angles of a triangle add up to 180 degrees, then space > containing the polygon has zero curvature, by the operative > definitions of zero-curving space. From where are you getting all these strange definitions? Are you just making them up on the fly to fit your original claim or? Perhaps you should qoute your sources, if you have any. In a flat space, the sum of the squares of the sides of a right angled triangle is equal to the square of the hypotenuse. This relationship does not hold for curved spaces. This means that if this is true for any right angled triangle in a given surface, then it is a flat space. If not, then it is a curved space. From this follows that a triangle in a curved space might have an angle sum of 180 degrees. You are making the mistake of saying that all birds are crows. When only the reverse is true, all crows are birds. Magnus
modified on Friday, April 16, 2010 7:05 AM
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Hello, you can draw this line, and you stated how: - Draw two perpendicular lines. - Take your compass and open it some width; it defines a unit. - With your compass measure, from the intersection of the two lines, measure one unit right and two units up. - Link the two resultant points with a line. This line measures SQRT(3). So you can effectively draw this line. Sum of angles of a triangle is 180 degree, yes, in an Euclidean geometry (plane geometry). But in other geometries this sum can be bigger, take this example: - Take the sphere as the surface of this geometry. - A "straight line" is the line that joins two points with the minimum length, in this case is a circle of maximum length. Over the sphere there are no paralel lines. - Consider the triangle defined by: the North Pole, the Ecuador at meridian 0, and the Ecuador at meridian 90. In this triangle all three angles measure 90 degrees. I cannot recommend you any bibliography, I'm not an expert on Geometry. But some articles about non-Euclidean Geometry in Wikipedia could be a good start in this theme. Regards, Francesc
Oops, of course, if one side is 1 and other side is 2 the hypotenuse measures SQRT(5), not SQRT(3). But there is not a problem; you can construct a triangle with sides 1 and SQRT(2) at the 90 degrees angle, and the hypotenuse will measure SQRT(3). Francesc
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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 assume 1, 3^1/2 and 2 are the lengths of the sides of the triangle and that you are using the Pythagorean theorem where a^2 + b^2 = c^2, The 3^1/2 (or SQRT(3))term seems fishy. If I were to let a = 1 and b = 2, then a^2 + b^2 = 1 + 4 = 5 = c^2. Then the third side should be 5^1/2 or SQRT(5). Would this help solve the question?
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Well since the square-root of 3 is a non-finite number, no you couldn't draw the line. Hence the figure drawn would not be a triangle at all since the two lines would never meet and the figure would not be closed. Ergo, the "point" were one side "doesn't meet" with the 3^1/2 side has no angle.
ahmed zahmed wrote:
Well since the square-root of 3 is a non-finite number, no you couldn't draw the line.
Sorry for my lateness to the conversation, but by that logic I couldn't draw a circle that has a integral radius.
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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.
Let me ad my 2 cents into how to draw line with length SQRT(3) units. 1. Draw square rectangle with sides equal to 1 unit. 2. Draw any diagonal of that rectangle. Accordingly to Pythagorean theorem length of diagonal will be SQRT(1^2 + 1^2) = SQRT(2) units. See http://en.wikipedia.org/wiki/Square\_root\_of\_2 Notice that we just draw line with length measured as irrational number SQRT(2) units. 3. Draw perpendicular with length 1 unit to the one end of the above diagonal. You can do it by using caliper. 4. Draw line from another end of diagonal to the free end of above perpendicular. Accordingly to Pythagorean theorem length of this line will be SQRT(SQRT(2)^2 + 1^2) = SQRT(3) units. Therefore, we draw line with length measured as irrational number SQRT(3) units. We also can do it using caliper and drawing circle and hexagon. See http://en.wikipedia.org/wiki/Square\_root\_of\_3
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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:
ChrisBraum wrote:
If you can draw a line of 1 unit accurately to .01% of the actual value, I have a job for you
I don't do nude portraits, sorry.
