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# Maths AA

What do the interior angles of a triangle add up to?
by

## Shawn Ong

on 21 April 2010

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#### Transcript of Maths AA

Mathematics Term 2 AA
By: Tan Yang Yi
Karl Png
Shawn Ong
The Big Question… What do the interior angles of a triangle add up to?
Firstly, we need to know the types of triangles and if the method we used will be applicable to all of them.
There are three types of triangles such as:
Isosceles Scalene Equilateral The most conventional method: The Cut Paste
This diagram shows the three interior angles of a triangle onastraight line. It proves that the interior angles of a triangle adds up to 180 degrees.
a b c a b c The next Method: Folding 1. 2. 3. 4. See? The angles make up a 180 degrees on a straight line! The next question: Do the interior angles of all triangles add up to 180 degrees? In Euclidean Geometry: If the pencil(writing tool) goes one round and back to the original point, why does the interior angles not add up to 360 degrees but instead 180 degrees? Well, firstly in a triangle it is impossible to have more than one angle at 90 degrees or more, thus at least two angles can be at a maximum of 60 degrees. It is these acute angles that leads to the interior angles of a triangle adding up to 180 degrees. As one angle widens another one becomes more acute at the exact same rate if you hold the third one, and if you don't hold the third one the sum of the change of the other two will equal the change of the first one. So to keep this proportionate the sum has to always be 180. In Riemannian Geometry : Interior angles of all Triangles do not necessarily add up to 180 degrees. This triangles are called ‘Non-planar triangles’. For example, longitudinal lines on the Earth originate at the North Pole. They intersect the Equator at 90 degree angles. The lines proceed to the North Pole and intersect there at whatever angle measure (x). 90+90+x=>180, assuming x>0, so in this form of Geometry, the triangles will always measure more than 180 degrees. This is possible as the lines drawn on the earth’s surface is not 2-dimensional but it is on a positively-curved surface. Similarly, triangles drawn on a negatively-curved surface will add up to a triangle with an interior angle of less than 180 degrees.
Simple Conclusion:
The interior angles of a triangle adds up to 180 degrees in Euclidean geometry(2-D) and the above explanations proves that the answer is the same for all triangles. Whereas in Riemannian Geometry(non-Euclidean), the answer differs. Thus it is absolutely up to what the reference is.
Credits:
1. http://math4allages.wordpress.com/2009/12/03/triangle-angle-sum/
2. http://en.wikipedia.org/wiki/Triangles#By_internal_angles