Triangle Angle Sum
The triangle angle sum theorem states that the three interior angles of any triangle always add up to exactly 180 degrees.
Formula
\text{angle } A + \text{angle } B + \text{angle } C = 180^\circ
Definition
The angles inside any triangle always add up to exactly $180^\circ$, a fact known as the Triangle Angle Sum Theorem and a direct consequence of the parallel postulate (proved by drawing a line through one vertex parallel to the opposite side and using alternate interior angles). If you know two angles of a triangle, the third is always $180^\circ$ minus their sum, and the same idea generalizes: the interior angle sum of any $n$-gon is $(n-2)\times180^\circ$, derived by dividing the polygon into $(n-2)$ triangles.
Example
Tearing off the three corners of a paper triangle and lining them up always forms a straight line; a triangle with angles $90^\circ$, $60^\circ$, and $30^\circ$ sums to $180^\circ$. To find the third angle of a triangle with angles $52^\circ$ and $73^\circ$: $180-52-73=55^\circ$. On a unit sphere, an octant triangle (three $90^\circ$ angles) has angle sum $270^\circ$, an excess of $\pi/2$ over $\pi$, matching its area of $\pi/2$ (one-eighth of the sphere) by Girard's theorem.
Key Insight
The $180^\circ$ sum is a signature of flat (Euclidean) space: on a sphere, triangle angles sum to more than $180^\circ$, and in hyperbolic space, to less, with the deviation from $180^\circ$ measuring the surface's curvature. This connects a simple classroom theorem to the geometry of the universe itself, since Einstein's general relativity uses this same non-Euclidean angle-sum deviation to define spacetime curvature, meaning a triangle drawn around the sun would have an angle sum slightly different from $180^\circ$.