Theorem 6.3: The sizes of the angles in a triangle always add up to   180o.

Proof:

Behold!

Given   triangle ABC,   by Theorem 1.7, there is a line through   C   parallel to the line determined by   AB.   Since alternate interior angles are the same size by Theorem 6.1, we can conclude that the sum of the sizes of angles in the triangle are equal to the sum of the sizes of the three angles meeting at point   C   at the top of the figure which can be seen to add up to a straight angle.

next theorem (6.4)