How this instrument works
The Segment Addition Postulate says that if point B lies directly between points A and C on the same straight line, the two smaller pieces add up to the whole: AB + BC = AC. Betweenness is the entire condition — B has to sit on segment AC itself, not merely somewhere near it, for the two shorter lengths to combine into a longer one that is exactly right rather than only close.
Unlike a theorem, a postulate is not proved from something simpler; it is adopted as a starting rule because it matches how a ruler behaves. Lay a number line through A, B, and C in that order and each point picks up a coordinate; the distance between any two points is just the difference of their coordinates, and subtracting shows the middle point's coordinate cancels out cleanly, leaving AB + BC = AC as an automatic consequence of ordinary subtraction rather than a fact needing its own separate proof.
Move B off the straight line joining A and C, even slightly, and the equality breaks down: AB + BC then overshoots AC, never falls short of it, which is the triangle inequality showing up. The Segment Addition Postulate is really that inequality's exact boundary case — equality holds only when the three points are collinear with B sandwiched in the middle, which is why a surveyor or carpenter treats a clean AB + BC = AC check as proof that three marked points actually lie in a straight line.
- Enter the first stretch's length into Length AB — the distance from point A to the point in between, B.
- Enter the second stretch's length into Length BC — the distance from B onward to point C.
- Read Length AC, the whole segment's length, computed as the sum of the two shorter pieces.
- Check that the result makes sense: Length AC should always come out larger than either Length AB or Length BC by itself, since B must sit between A and C for the postulate to hold.
Worked example — a fence line split at post B
A straight fence runs from corner post A to corner post C, with a third post B planted somewhere along the run to break the span up for bracing. A surveyor measures Length AB at 5 metres and Length BC at 7 metres. Because B sits directly on the line joining A and C, the Segment Addition Postulate applies without qualification: AC = AB + BC = 5 + 7 = 12 metres, the full fence line measured in two short stretches instead of one long one.
The postulate also works as a check on the survey itself. If a separate, single measurement of the whole fence line had come back as anything other than 12 metres — say 11.5 — that mismatch would flag either a measuring error at one of the three posts or, more interestingly, that B is not actually sitting on the straight line between A and C, since only a truly collinear middle post produces an exact sum.
Questions
What is the Segment Addition Postulate?
If point B lies between points A and C on the same straight line, then AB + BC = AC — the length of the whole segment equals the sum of its two parts. It is one of geometry's foundational postulates: accepted as a starting rule rather than proved from something simpler, because it matches exactly how distances behave along a number line.
Why is this called a postulate instead of a theorem?
A postulate is a statement taken as a starting rule rather than derived from earlier results, the way Euclid built his geometry from a short list of them. This one earns that status because it follows directly from how points get assigned coordinates on a line: once coordinates are fixed, AB + BC = AC drops out of ordinary subtraction, so stating it as a rule saves reproving that same subtraction every time it comes up.
Does AB + BC always equal AC?
Only when B lies exactly on the straight line between A and C. Move B off that line, even slightly, and AB + BC becomes strictly greater than AC instead of equal to it — the triangle inequality at work. The equality here is the special, collinear boundary case of that broader inequality, not a separate rule of its own.
How is this postulate used inside a geometry proof?
It usually justifies a substitution step: once a diagram or a coordinate check establishes that B lies between A and C, a proof can freely write AC as AB + BC, or solve that equation for either smaller piece, without further argument. Two-column proofs typically cite it by name right after betweenness has been established.
How does this relate to the Angle Addition Postulate?
They apply the same idea to different measurements. The Angle Addition Postulate says that if a ray splits an angle into two smaller angles, those two measures sum to the whole angle, exactly as adjacent segment lengths sum to the whole segment here. Betweenness plays the identical role in both: a ray strictly inside the angle, a point strictly on the segment.
Can this postulate find a missing middle segment instead?
Yes — rearranged, it isolates either smaller piece: AB = AC − BC, or BC = AC − AB. Given the whole segment's length and one of its parts, subtracting recovers the other part directly, the same relationship this calculator runs in its addition direction rather than its subtraction direction.