Why Do the angles of all triangles equal a 180 degrees?

Answer 1

Here is the proof, it is done by drawing parallel lines and using the theorem that alternate interior angles of parallel lines are congruent, which is also easily proved: The Pythagorean proof is even simpler. A line parallel to the base BC is drawn through the vertex A, which gets us two pairs of alternate angles: CBA, DAB and BCA, EAC. Now, angle BAC complements, on the one hand, CBA + BCA to the sum of the angles of DABC, and, on the other, DAB + EAC to two right angles. Basically this says that because a straight line has 180 degrees on each side, and that angles B and C are both alternate interior angles to angles BAD and CAE respectively, angles B and C can both be substituted by angles BAD and CAE. Therefore, you get angle C + angle B + angle A = 180.