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Proof the sum of the interior angles of any triangle is always 180 degrees or Pi.


Firstly, we need to know about a measure of angles called radians.  There are, for example, PI (3.14159...) radians in 180 degrees.



                                                                   Animation provided by Lucas V. Barbosa.



Furthermore, here are some basic relationships between lines and angles...


Draw two lines that intersect with each other and then label the diagram where P and Q are both angles.  Notice by symmetry, the following is true.



We do not know what the two angles P and Q are, but one very important thing we do know is:

Where P and Q, and PI are all measured in radians.



Now take one of the lines (it doesn't matter which one) and draw a line parallel to it like so in green.





Because the green line is parallel to the other line, the following relationships between the angles can be shown as follows...




That's pretty much all you need to know for the proof but some annotation of a triangle is appropriate.



So take any triangle and extend the lines as shown and label the angles of the triangle formed A, B, C. 



Now pick any of the three vertices (it doesn't matter which one).  Let's take vertex A and label the other angle as E say.





Next, draw a line through vertex A such that it is parallel to the opposite side of angle A 



This means the following is true,



 We don't need E anymore and similarly the following is true.