Question

1. line segment cd is the perpendicular bisector of line segment ab. is line segment ab the perpendicular bisector of line segment cd?

Explanation:

Step1: Recall perpendicular bisector definition

A perpendicular bisector of a line - segment divides the segment into two equal - length parts and is perpendicular to it. Just because $CD$ is the perpendicular bisector of $AB$ (i.e., $AM = MB$ and $CD\perp AB$), we have no information about the lengths of $CM$ and $MD$ or whether $AB$ is perpendicular to $CD$ at the intersection point.
A counter - example can be constructed where $CD$ is the perpendicular bisector of $AB$, but $AB$ does not bisect $CD$ (for example, when $A$ and $B$ are close to the mid - point of $CD$ but $CD$ is much longer than $AB$ and not symmetrically placed with respect to $AB$).

Answer:

No