what did we use: polydrons, polydron protractor

J2 and I were playing with polydrons and he got interested in measuring the various angles. Since there are three different triangles in the set, he got to test each of them and found that the angle sum was always 180 degrees. We went on to examine squares and pentagons to see what he could make of those.

In the evening, P and I were talking about about this investigation and I mentioned this proof that the interior angles of a polygon sum to $180 \times (n-2)$:

P was surprised and said that she likes this proof more:

In this version, you have to subtract the middle 360 degrees.

I was a bit surprised that I hadn't ever thought of this version or seen it elsewhere, but quickly realized we could repurpose it to finding the interior angles of a triangle itself. Here's the picture:

Now, the argument:

However, our proof doesn't seem to use any fancy postulates. Your challenge, why doesn't it work on a sphere? What extra postulate did we secretly use?

P was surprised and said that she likes this proof more:

In this version, you have to subtract the middle 360 degrees.

I was a bit surprised that I hadn't ever thought of this version or seen it elsewhere, but quickly realized we could repurpose it to finding the interior angles of a triangle itself. Here's the picture:

Now, the argument:

- Angles of the large triangle are equal to the sum of the angles of the smaller triangles - 360
- Let x be the sum of interior angles of a triangle.
- Based on 1, we have x = 3x - 360
- Rearranging, we get x = 180.

# Where we get stuck: a challenge for you

Here's the thing: the sum of interior angle measures of a triangle*180 degrees. That is, it doesn't have to be if you are working in non-euclidean geometry. Here's a nice picture (not mine) of a triangle with angle sum of 270 degrees:***isn't**Comes from MathStackExchange |

However, our proof doesn't seem to use any fancy postulates. Your challenge, why doesn't it work on a sphere? What extra postulate did we secretly use?

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