The Triangle Angle Sum Theorem

The rule is simple: angle one plus angle two plus angle three equals 180 degrees, no matter how the triangle is shaped. You can see why by tearing off the three corners and placing them together, point to point, along one line. The three torn corners fit edge to edge and form a straight line, which measures exactly 180 degrees. That is the proof, and it works for every triangle, thin or fat, right or obtuse.

On the test this shows up three ways. Plain version: you're given two angles and solve for the third by subtracting their sum from 180. Ratio version: the three angles split 180 degrees into parts, so you add the ratio numbers, divide 180 by that total to get the value of one part, then multiply to find the angle you want. Exterior angle version: an exterior angle and its adjacent interior angle sit on a straight line and sum to 180, while that same interior angle plus the other two interior angles also sum to 180 — so the exterior angle always equals the sum of the two far corners, no subtraction needed.

This theorem sits underneath almost every other triangle question on the test, including right-triangle trig and similar triangles, because knowing two angles always hands you the third for free. Spot a triangle anywhere in a figure, even hidden inside a polygon or circle, and this is the first fact to reach for.