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°.

5 Postulates of Euclidean Geometry

1. "To draw a straight line from any point to any point."

2. To produce [extend] a finite straight line continuously in a straight line."

3. "To describe a circle with any centre and distance [radius]."

4. "That all right angles are equal to one another."

( right angle = 90˚ )

5. The parallel postulate: 

"That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, 

if produced indefinitely, meet on that side on which are the angles less than the two right angles."

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