Euclidean Foundations: Reflections, and Orthogonality of x- and y-Axes

There's one missing piece of structure in our ordered pair definition of points.

It boils down to this: What's the relation between the $x$- and $y$-axes?

We call this fundamental relation orthogonality (or perpendicularity). The $x$- and $y$-axes are orthogonal to each other!

A fundamental transformation related to orthogonality is the reflection. The reflection over the $x$-axis takes $(a,b)$ to $(a,-b)$, negating the $y$-coordinate. Similarly, the reflection over the $y$-axis takes $(a,b)$ to $(-a,b)$.