The Segment Addition Postulate

When three points lie on a line, the middle one splits the segment into two pieces — and those pieces add up to the whole. That is the Segment Addition Postulate: if $B$ is between $A$ and $C$ , then $AB + BC = AC$ . It looks obvious, and that is exactly why it is useful: it turns a picture of a line into an equation you can work with.

Once you have equations, the ordinary properties of equality — adding the same length to both sides, subtracting it away, substituting equals for equals — do the rest. A relationship you could only see becomes one you can prove.

The idea

Line up four points $A$ , $B$ , $C$ , $D$ in order. The Segment Addition Postulate gives $AB + BC = AC$ and $BC + CD = BD$ : each overlapping segment is its two pieces. If the two outer pieces are congruent to begin with, then adding the shared middle piece $BC$ to both makes the whole segments equal — so $AC \cong BD$ . Every step is a named reason, not a leap.

Build the proof

Read the Given and Prove, then supply the reason that justifies each statement. Click a segment in the figure to find where it appears in the proof, or click a statement to see the segment it is about. Only a step that genuinely follows from the ones above it will lock into place.

Given: $\overline{AB} \cong \overline{CD}$

Prove: $\overline{AC} \cong \overline{BD}$

Statements	Reasons
1.
2.
3.
4.
5.
6.
7.
8.

Row 1: choose the reason that justifies this statement.

Reasons

Placed 0 of 8 steps.

Points sit on a line, with congruent segments tick-marked. Build a two-column proof that

\overline{AC} \cong \overline{BD}

: each statement is given in order and you supply the reason that justifies it. Each row is checked against the logic of the proof, so only a step that genuinely follows will seat.

Key result

The Segment Addition Postulate converts betweenness into an equation ( $AB + BC = AC$ ). From there, congruent segments become equal lengths (definition of congruent segments), the properties of equality combine or cancel the shared piece, and substitution swaps a sum for the segment it equals — turning a figure you can see into a conclusion you can prove.

For teachers

Standard, common misconceptions, and suggested use will live here.