The Triangle Sum Rule is a theorem from Euclidean Geometry that states:
"In any triangle, the sum of its three angles is 180 degrees."
This theorem is a direct result of the Parallel Postulate and is proved as follows:
Draw triangle ABC. Through point C, construct a line parallel to AB. We know such a line exists (and is unique) by the parallel postulate. We label two points on this line H and K for help in naming angles later:
Now, since HK is parallel to AB, this means that angle HCA is congruent to angle A, since they are alternate interior angles. Likewise, angle BCK is congruent to angle B.
Since HCA, ACB, and BCK together form a straight angle, this means:
HCA + ACB + BCK = 180 degrees
By the congruencies above, we can use substitution to obtain:
A + ACB + B = 180 degrees
Note that these three angles are the angles of triangle ABC. Hence, the three angles of an arbitrary triangle must add up to 180 degrees. QED.
As mentioned above, the triangle sum rule is a direct result of the parallel postulate. In fact, the triangle sum rule is equivalent to the parallel postulate, that is, if we removed the parallel postulate from the axiomatic system that comprises Euclidean geometry, and secretly replaced it with the triangle sum rule, the resulting geometries would be identical (except, of course, we'd have to prove what we know as the parallel postulate from the triangle sum rule, instead of vice versa).
In non-Euclidean geometry, in general, the triangle sum rule does not hold, since the parallel postulate is not assumed. In the "standard" non-Euclidean geometries (spherical and hyperbolic), the following modifications of the triangle sum rule apply:
In spherical geometry, the sum of the angles of a triangle exceeds 180 degrees in an amount directly proportional to the triangle's area. The difference between 180 degrees and the triangle's angle sum is called the triangle's defect.
In hyperbolic geometry, the sum of the angles of a triangle is less than 180 degrees, again in an amount directly proportional to the triangle's area. (In fact, in some implementations of hyperbolic geometry, the defect of the triangle in radians is equal to the triangle's area.)