Graphical proof that the sum of angles in a triangle is 180deg.


Consider the following diagram:

1.  Start with the vertical vector along the line that connects angles a and c.  Rotate counter-clockwise* until the vector is parallel with the side connecting angles b and a.  The vector has now been rotated through an angle equal to a.

2. Now rotate the vector through an angle equal to b, making it parallel to the side connecting angles b and c.

3.  Finally, rotate the vector through an angle equal to c, so that it will be pointing in exactly the opposite direction from its original position.




*  "counter-clockwise": An ancient term which dates back to a time when clocks had rotating pointers called "hands".  There were two hands, one long and one short.  The short hand had the longest period, (12 hours); The long hand had the shorter period, (1 hour).  (Some clocks had a third hand called a "second" hand which had a period of one minute.)  By convention, the tips of the hands would be moving to the right when the hands were pointing straight up - this defined clockwise rotation.  Counter-clockwise is rotation in the opposite direction.

**  The hands of the clock, by the way, rotated in front of a circular scale on a fixed plane surface called the "face".  The circular scale was divided into twelve major divisions, and each major division was divided into five minor divisions.  Be thankful that you live in the digital age.




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