Thought Question for Today

[Gregory Matthews]

Back in ancient times when you studied geometry you learned that in a triangle the sum of its  interior angles always added up to 180 degrees.

However, we now know that such does not always happen.  The following are examples that may actually happen:

*  The sum of the interior angles of a triangle may add up to more than 180 degrees.

*  The sum of the interior angles of a triangle may add up to less than 180 degrees.

 

As a thought question for today, under what circumstances may each of the above be true?

[Gregory Matthews]

Yes, I posed my question in terms of Euclidean geometry because that is what you were taught in high-school.

To start you to think, see if you can conceptualize a triangle with two (2) interior angles of 90 degrees each and one interior angle of 179 degrees.  Yes, such a shape could exist.  Also so could a triangle with two interior angles of 90 degrees each and one interior angle of 17 degrees.   In order to conceptualize those shapes one must think outside of the box.  In doing so, you have began to think about triangles formed on the surface of a sphere.

 

[Gregory Matthews]

Wanderer, one of the keys to understanding this is  to understand that Euclidean geometry involves a 2-dimensional, flat space of a plane.  Spherical geometry involves a 3-dimensional shape.

 

[Gregory Matthews]

I debated with myself between you and another person as to who would be the first to respond to me.  In the end, I decided that you would be the first and possibly the other person would not respond at all.