[Math] When we have two line segments $AB$ and $CD$, what does $AB=CD$ mean

Question

Suppose that we have two line segments, AB and CD. We know that they have the same length.

I know that $\overline{AB}=\overline{CD}$ means $AB$ is identical to CD (aka. They are the same lines), and also that $\overline{AB}\cong\overline{CD}$ means that $AB$ and $CD$ have the same size, but what does $AB=CD$ mean?

I actually saw this in a proof of the Transitive Property of Congruence. This is the proof:

Answer 1

In the horrible, no-good world of two-column proofs:

• $\overline{AB}$ and $\overline{CD}$ are line segments.
• $AB$ and $CD$ are the lengths of line segments $\overline{AB}$ and $\overline{CD}$.
• $AB = CD$ just means that the two positive real numbers which are the lengths of those line segments are equal to each other.
• $\overline{AB} \cong \overline{CD}$ means that the two line segments are congruent, where we define two line segments to be congruent if and only if they have the same length. (That is, we define the relation $\cong$ so that $\overline{AB} \cong \overline{CD}$ if and only if $AB = CD$.)

There are maybe some reasons to make the distinction between congruence of line segments and equality of lengths. But since nobody in high school geometry classes ever talks about those reasons, this is just an exercise in dealing with dumb definitions.