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

Question

I have been told there is a time when the angles in a triangle add up to more than 180 and another occasion where it adds up to less than 180? How and when?

i found it out: when it is on the outside of a sphere it is equal to slightly more and when it is on the inside it is slightly less.

Answer 1

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 2

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 3

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 4

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 5

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 6

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 7

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 8

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 9

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 10

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