Proof of Sum of Angles of A Triangle

Before you see this post,

see

5 Postulates of Euclidean Geometry 

[Pic. 1]

Use The parallel postulate, we can find those facts in [Pic. 1]

[Pic. 2]

There is a triangle.

0°<{α,β,γ}<180°

[Pic. 3]

We can draw like [Pic. 3] by

 2nd of  5 Postulates of Euclidean Geometry.

[Pic. 4]

let's draw l₄ is parallel with l_2.

[Pic. 5]

We can find sum of angles of the triangle by

facts in [Pic. 1].

α+β+γ = 180°

So, sum of angles of the triangle is 180°.