The formula
A + B + C = 180°
Where it comes from
The three angles of any flat triangle add to 180°. The proof is a single line: draw a line through one vertex parallel to the opposite side, and the three angles reappear side by side along it, making a straight angle.
How to work it out by hand
- Add the angles you know
- Subtract from 180°
- If the result is zero or negative, no such triangle exists
- Classify: all angles under 90° is acute, one at 90° is right, one over 90° is obtuse
What is worth knowing
The 180° rule holds only on a flat surface, and that is not a technicality. On a sphere the angles of a triangle add to more than 180°, and the excess is proportional to its area — a triangle covering an eighth of the Earth has three right angles and adds to 270°. That is why long-haul navigation cannot use plane trigonometry, and it is the entry point to non-Euclidean geometry: dropping the parallel postulate is exactly what changes the angle sum.