the vertex angle sum of a triangle is 180 degrees. why?

Answer 1

Because a triangle has only three sides and all sides must end touching another side.

Shapes with different sums are not triangles. They have to have more than three sides to go more than 180

Answer 2

The proof is simple. Draw a line || to base and through the vertex opposite the base. Then using the fact that alternate interior angles formed by a transversal cutting two parallel lines are equal, you can quickly show that the vertex angle plus the two base angles add up to a straight angle (the || line) and is thus 180 degrees.

Answer 3

Triangles are intrinsically drawn by a tiny invisible turtle. This turtle is very tiny, So tiny indeed that you can't see it. Plus it's invisible.

As the turtle crawls around the triangle (which it does as fast as the pencil traces over) it needs to make the turns for the corners. Once it completes all turns it's gone full circle making the entire traversal 360 degrees.

But, because only two legs of the turtle are on the inside of the triangle, we only count half of those degrees, hence 180

-arr

Answer 4

Because it is twice 90 degrees.

Answer 5

Here is an excellent graphical proof that is very simple.

http://www.mathsisfun.com/proof180deg.html