How can the angles in a triangle add up to less than 180 degrees?

Question

How can the angles in a triangle add up to less than 180 degrees?

Answer 1

On the surface of a sphere it's possible to have a triangle with more than 180 degrees. In plane, or 2 dimensional Euclidean geometry, triangles have 180 degrees. You might want to think about a degenerate triangle where you take a right triangle and shrink one of the legs down to zero length. You have one angle approaching zero, and two angles approaching 90, so it still works.

Answer 2

The said triangles can be produced using non Euclidean geometry.



The word triangle (tri-angle) just means a closed shape with three angles. It is not compulsory for the lines to be straight for the closed shape to be a triangle.



99.9999999999999% of the time when we see triangles the lines are straight, but they don't have to be.

Answer 3

it's impossible-the angles in a triangle

must equal 180 degrees in

plane geometry



proof:



if you draw a triangle between two parallel

lines,it can be shown quite easily

that the angles of a triangle add up to

180 degrees-this is on a plane surface not

a curved surface though



i hope that this helps

Answer 4

Only in spherical geometry, not in planer geometry.

Answer 5

It is not possible

Answer 6

It can't. A triangle's angles always add uo to 180 degrees.

Answer 7

I was always sure they couldn't, but if they could it probably wouldn't be a closed figure, like the symbol: [ or ]

Answer 8

They are lying, it is impossible



ALL ANGLES ADD UP TO 180!

Answer 9

doesn't happen, unless the triangle is a square!

Answer 10

YES - in non-planar geometry e.g. spherical trigonometry.

This is one example of a non-Euclidean geometry. (Hyperbolic geometry would be another)



A triangle on the outside of a sphere always has >180deg



"Remarkably, the sum of the vertex angles of a spherical triangle is always larger than the 180° found in every planar triangle. The amount by which the sum of the angles exceeds 180° is called the spherical excess E: E = α + β + γ − 180°. This surplus determines the surface area of any spherical triangle. To determine this, the spherical excess must be expressed in radians; the surface area A is then given in terms of the sphere's radius R by the expression:



A = R^2 · E. From this formula, which is an application of the Gauss-Bonnet theorem, it becomes obvious that there are no similar triangles (triangles with equal angles but different side lengths and area) on a sphere."



Spherical law of cosines:

cos c = cos a cos b + sin a sin b cos C



(A spherical triangle abc is specified as usual by its corner angles a,b,c and its sides A,B,C, but the sides A,B,C are given not by their length, but by their arc angle.)



http://en.wikipedia.org/wiki/Spherical_trigonometry#Identities

Answer 11

Do the angles of a triangle add up to 180 degrees or p radians? The answer is 'sometimes yes, sometimes no'. Is this an important question? Yes because it leads to an understanding that there are different geometries based on different axioms or 'rules of the game of geometry'. Is it a meaningful question? Well no, at least not until we have agreed on the meaning of the words 'angle' and 'triangle', not until we know the rules of the game. In this article we briefly discuss the underlying axioms and give a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is not equal to p but to p plus the area of the triangle. We shall use the fact that the area of the surface of a unit sphere is 4p.

Answer 12

If you draw what looks like a triangle on a sphere when looking at it from a distance, the actual shape you draw on the curved surface is not a triangle. It only looks like it because of how you are looking at it and seeing it as an assumed flat surface which it is not. cut a triangle out of some thin paper, check that the angles add up to 180 degrees. now place the triangle on a sphere i.e. a ball etc, now look at the triangle, does it still look like a triangle, you should see that it looks distorted (especially the corners) but you know that it really is a triangle.

Answer 13

The angles in a triangle can't add up more than 180 degrees. Think of it like this... if you have a right triangle, the largest degree in that triangle is 90 degrees. Now, skew the triangle, just increasing the largest degree (90 deg). If that angle keeps getting larger, approaching a straight line, the other two angles are approaching zero degrees. So if you get to zero, you've got a straight line, which is 180 degrees from 0 to 180 degrees. The reverse applies for the angles adding up to less than 180 degrees.

Answer 14

Imagine this triangle on the surface of a sphere. One line from the North Pole through London to the equator. Another line from the North Pole through Los Angeles to the equator. And finally a line along the equator connecting the other two.



The angle at the pole is way larger than 90 degrees. The angles at the equator are both 90 degrees. A triangle with more than 270 degrees!!!

Answer 15

This is all Euclid's fault! His fifth postulate states that thru any point not on the given line there is a unique line that can be drawn parallel to this given line. Sounds intuitive, but not obvious. Can't be proven from preceding axioms tho'. Solution(s): assume 5th postulate is false-does this lead to contradictions involving previous 4? Apparently not! So why not postulate a geometry system in which 1) no parallel lines exist. 2) an infinite No. of parallel lines exist? ('concave' and 'convex' spaces respectively) Both are equally valid. No one thought of them as no more than theoretical abstractions till Einstein came along with his theory of general relativity and space time curvature. Basically if you lay out a triangle in a gravitational field the sum of the angles< 180deg 'Matter tells space(time) how to curve, Space(time) tells matter how to move'! Basically many of the concepts involved in SPECIAL relativity look paradoxical or counter-intuitive (like time dilation etc) but make perfect sence when viewed from the point of view of 4D space-time geometry!

Answer 16

You are having your leg pulled, all angles in a triangle add up to 180 degrees, anything else and it is not a triangle.

Answer 17

With the Bermuda Triangle. Eerie things happen there, apparently. Or a love triangle. Otherwise, it's 180 degrees.

Answer 18

Unless that the lines are not straight then that is impossible but as everyone knows ALL STRAIGHT LINES IN A TRIANGLE ADD UP TO 180 DEGREES.

Answer 19

That is not true if the interior angles do not add up to 180 it is not a triangle.

Answer 20

If the triangle is a triangle then no. The 180 (o) rule you wrote is one of the things that make a shape a triangle.

Answer 21

when i triangle enters the last phase of its two dimensional being there can be a tremor around the edges which to the untrained eye can appear to be in flux between 179 and 181 degrees but here at curtlabs we have proved absolutely that this is just an optical phenomena.

squares however tend to angle shift at much higher occilation and therefore do not adhere to the same pseudo-stratifications.

Answer 22

only if the lines arent straight. Any 3 straight lines that join each other will have angles equal to 180

Answer 23

If the triangle has 4 sides!!!!

Answer 24

as far as i know, that isnt possible. unless it was not made out of straight lines, but then it wouldnt be a triangle.

Answer 25

are you sure you weren't having your leg pulled? I'm sure I remember being told it could only be 180%

Answer 26

there is no way this can happen in eucledian geometry. u've either gotten the wrong info or you're talking about something really really complex.

Answer 27

if it didn't it would not be a triangle if it did the point would not match up?

Answer 28

iN ANY TRANGLE THE THREE ANGLE THA T MAKE UP THE TRIANGLE IS IT IT IS IMPOSSIBLE AS IMPOSSIBLE AS IT OR AS IF ANYTHING CAN BE ASBOLUTELY IMPOSSIBTLE FOR THE THREE ANGLE OF asNY AND I REPEAT aNY TRIANGLE TO BE LESS THAN 180 DEGREEE THE PERSON THAT TOLD YOU THIS IS A DAMN LAIR !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer 29

Maybe they need a new calculator

Answer 30

I think it is impossible. It was when I was at school.

Answer 31

Dunno. Maybe they're rounding up or down. Like if one angle is 74.9 degrees and they round up to 75. Or rounding 74.3 to 74...?

Answer 32

i didnt think it could, i thought it impossible.....let me know if you find out it does...