How to prove that angles on straight line add up to 180 degrees?

Question

Hi
I want to know how to prove that angles on straight line add up to 180 degrees, WITHOUT using theorems that their proofs use the theorem i want to prove, and without using theroems that their proofs contain use of another theorem, and this theorem proof contains the fact i want to prove.
I think you got what i mean, i want to know how people figured this out, and of course they did not have theorems that sum of angles on straight line implies - because they didn't know it yet.
what i tried to do, is take a straight line AB, take a point O on AB such that AO=BO=R, and build a circle with center O and radius R.
http://sketchtoy.com/68586942 look here what i mean and look at the angles named alpha and beta (sorry for the bad drawing)
I can see that alpha+beta=360 because of the defenition of degree.
but how can I know that alpha=beta?
if i can prove that the diameter AB cuts the circle to two equale arcs, i can know that alpha=beta and then alpha=beta=180, it is very intuitive that this is the case but how can i prove it? thanks!

Answer 1

but how can I know that alpha=beta?

$\alpha=\beta$ because your figure stays the same when you mirror it w.r.t. the line.

Answer 2

When you draw a circle whose center is on your straight line the circle is divided into two equal semi-circles.

The total degree of a circle is by definition $360$ degrees.

Thus each semi circle accounts for $180$ degrees.

That is the central angle is also $180$ degrees.

Thus the straight line add up to $180$ degrees.