the triangle angle-sum theorem
Tear the three corners off any paper triangle and lay them snugly side by side at a single point: they fit together to make a straight line, every time. That is the triangle angle-sum theorem in your hands — the three interior angles of a triangle always add up to 180 degrees, no matter the triangle's shape or size.
Precisely: for any triangle ABC, m(angle A) + m(angle B) + m(angle C) = 180. The classic proof draws, through vertex A, a line parallel to the opposite side BC. The two base angles B and C become alternate interior angles with that parallel line, so copies of them appear at A, on either side of angle A. The three angles at A now sit along a straight line, so their measures add to 180 — and those three are exactly the three angles of the triangle. The whole proof rests on the parallel-line angle theorems, so it is genuinely Euclidean.
This 180 is special to flat (Euclidean) geometry. On a sphere the angles of a triangle add to more than 180, and in the hyperbolic plane they add to less; the gap from 180 is called the angle excess or angle defect and measures the curvature. So 'angles sum to 180' is not a universal law of all triangles everywhere — it is a hallmark of the flat plane, and a beautiful early sign that the parallel postulate is doing real work.
A triangle has angles of 90 and 35 degrees. The third angle must be 180 - 90 - 35 = 55 degrees, since the three interior angles always total 180.
Two angles fix the third; all three sum to 180.
The 180-degree sum holds in the flat plane only. On curved surfaces the sum differs, and that difference is precisely curvature — so this theorem quietly depends on the parallel postulate.