the ruler postulate
Lay a ruler along an edge and you can name each spot by the number it touches, and the length between two spots is the difference of those numbers. The ruler postulate is the formal promise that this everyday move always works: every line can be turned into a number line.
The postulate states that the points of any line can be put into one-to-one correspondence with the real numbers, so that each point P is matched with a real number called its coordinate, and the distance between two points equals the absolute value of the difference of their coordinates: if A has coordinate a and B has coordinate b, then AB = |a - b|. You are free to choose where 0 sits and which direction counts as positive; the distances come out the same either way. In effect, this postulate hands you a tape measure for every line.
Small as it looks, the ruler postulate is the foundation that lets us measure at all. From it follow the meaning of length, of betweenness (B is between A and C exactly when AB + BC = AC), of the midpoint (the point at the average coordinate), and of segment congruence. Euclid had no such explicit postulate — his Elements treated length geometrically and leaned on diagrams — and supplying a measurement axiom like this is part of how later mathematicians, especially Birkhoff and the school-geometry tradition, made the foundations watertight.
On a line let A, B, C have coordinates -3, 1, 6. Then AB = |1 - (-3)| = 4, BC = |6 - 1| = 5, and AC = |6 - (-3)| = 9. Since 4 + 5 = 9, B lies between A and C, all read straight off the coordinates the ruler postulate guarantees.
Coordinates on a line turn distance into subtraction: AB = |a - b|.
Euclid had no explicit measurement postulate; the ruler postulate is a modern addition (Birkhoff, school geometry) that makes length and distance rigorous from the start.