the segment addition postulate
Walk from home to school by way of the corner shop, and the distance home-to-shop plus shop-to-school equals the whole distance home-to-school — provided the shop is on the way. The segment addition postulate says exactly this for points on a line.
The postulate states: if point B lies between points A and C, then AB + BC = AC. The two short lengths add up to the long one whenever the middle point really is in between. It works in reverse too, as a test for betweenness: if AB + BC = AC, then B must lie on segment AC between A and C. The condition that B is between A and C is essential — if B were off to the side or beyond C, the equation would fail.
This postulate is the workhorse that lets you split a segment, find an unknown length, and run length-based proofs. To find a missing piece you just subtract: if AC = 20 and AB = 13 with B between A and C, then BC = AC - AB = 7. It is the segment cousin of the angle addition postulate, and together with the ruler postulate it underpins the definition of the midpoint and every two-column proof that adds or subtracts lengths.
B is between A and C with AB = 3x and BC = x + 4, and the whole AC = 24. Then 3x + (x + 4) = 24, so 4x = 20 and x = 5; hence AB = 15 and BC = 9, and indeed 15 + 9 = 24.
When B is between A and C, the part-lengths sum to the whole: AB + BC = AC.
The equation AB + BC = AC holds only when B is between A and C; with B off the segment the lengths do not add this way. The 'betweenness' clause is doing real work.