If the student attends a school that has a distinct Geometry course (typically taken in 9th or 10th grade) then it will probably use a textbook similar to Glencoe Geometry or HMH Geometry. I have one of those open in front of me now; looking at it, I see that the sequence of topics runs like this:
- First, "perpendicular lines" are defined as lines that intersect to form 90° angles.
- Second, "a line perpendicular to a plane" is defined to mean a line that is perpendicular to every line in the indicated plane.
- "Parallel planes" are defined, and one of the postulates in the book is that if two planes are non-parallel then they intersect in a single line.
But (somewhat to my surprise) "two perpendicular planes" does not appear to be defined or ever used anywhere in the context of the book. I don't know that it's possible to say why that concept is never invoked, but I have a conjecture: The typical (?) definition of "perpendicular planes" goes like this:
A planes $M$ is perpendicular to a plane $N$ if there exists some line $m$ in $M$ that it perpendicular to every line $n$ in $N$.
That, at least, is the definition I found in the first few Google search results for "perpendicular planes definition" (examples: here and here). Notice, first, that this definition involves two quantifiers: An existential quantifier on $m$ (we only need to find one line $m$ in $M$ that meets the required condition), and a universal quantifier on $n$ (once we have chosen $m$, every line $n$ in $N$ needs to be perpendicular to it). That's conceptually quite subtle, arguably on a par with the well-known conceptual difficulties around the quantification in epsilon-delta proofs.
Second, notice that although the property being defined is symmetric (i.e. "$M$ is perpendicular to $N$" means exactly the same thing as "$N$ is perpendicular to $M$"), the definition is not. In fact it is not at all obvious that the existence of an $m$ in $M$ that is perpendicular to every $n$ in $N$ implies the existence of a specific $n$ in $N$ that is perpendicular to every $m$ in $M$.
I suspect that the conventional wisdom among curriculum designers, to the extent that it has given any thought to the matter at all, is that the potential benefit of defining perpendicular planes is outweighed by the difficulties entailed by the definition.
It seems to me that another possible definition would be to say:
Suppose planes $M$ and $N$ intersect in line $l$, and choose any point $P$ on $l$. Let $m$ be the line in $M$ that is perpendicular to $l$ at $P$, and let $n$ be the line in $N$ that is perpendicular to $l$ at $P$. Then planes $M$ and $N$ are called perpendicular if the lines $m$ and $n$ are perpendicular.
This definition has the advantage of being obviously symmetric in $M$ and $N$. It also generalizes, with very little work, to define a dihedral angle: The angle between $M$ and $N$ is defined to be the angle between $m$ and $n$. With this definition, $M$ is perpendicular to $N$ if the dihedral angle between them is 90°.
But whatever advantages this other definition may have, they are also probably outweighted by the semantic complexity of the definition, which requires six geometric objects (not only the two planes $M$ and $N$ but also the three lines $l, m$ and $n$ and the point $P$) to define a single relationship. For 9th or 10th graders, it's a disaster waiting to happen.
So it does not surprise me that Geometry textbooks avoid the problem of defining what it means for two planes to be perpendicular. On the other hand it does surprise me that they avoid the concept altogether. High school geometry typically treats rigor as a kind of aspirational value, not an absolute commitment. I think it's pretty surprising that I don't find any exercises that show, for example, a three-dimensional solid (say a frustrum) and ask the student to identify which planes are parallel, which are perpendicular, and which are neither.
Now, all of the above assumes you are interested in the geometry of lines and planes in space from a synthetic or Euclidean perspective. But if you want "parametric and symmetric equations" for planes (which I see was part of your question), then a Geometry course is the wrong place to look -- you want to look at a Precalc book. I have one in front of me: It's the Glencoe Advanced Mathematical Concepts: Precalculus with Applications. In this textbook we find that:
- An equation of the form $ax+bc+cz=d$ defines a plane in 3-dimensional space
- So a system consisting of two such equations defines two planes, which either are parallel (in which case there is no solution), or intersect in a single line (in which case there are infinitely many solutions that can all be expressed in terms of a single parameter), or coincide (in which case there are infinitely many solutions that can all be expressed in terms of two parameters)
- And a system consisting of three such equations defines three planes, which either do not intersect, or intersect in a line, or intersect in a single point, or coincide.
A typical treatment looks something like this:
But notice that in this context -- solving systems of equations in three variables -- it does not matter whether the planes are perpendicular or not. All that matters is whether they intersect; the angle of intersection is irrelevant.
Much later in a Precalc book, we encounter information about vectors, dot products, and cross products. In this context students learn that two vectors are perpendicular if and only if their dot product is zero. They also learn that the cross product of two vectors is perpendicular to the plane in which those vectors lie:
Such properties make it possible for students to answer questions like:
- Given two vectors, determine if they are perpendicular
- Given three points in space, determine if any of the vectors joining those points are perpendicular to each other
- Given two vectors, find a third vector perpendicular to both
- Given three points in space, find a vector perpendicular to the plane containing the three points
Students also learn to:
- Given a point $P$ and a vector $\vec{v}$, write a pair of equations for the line through $P$ in the direction of $\vec{v}$ in parametric form
- and vice versa.
This all gets pretty close to what you are asking for. But I still don't see in this textbook any place where students encounter the definition of a normal vector to a plane, or study the relationship between the equation of a plane and its orientation in space, or talk about what it means for two planes to be perpendicular. I think that material is simply beyond the scope of high school mathematics in the United States.
One more thought: You asked about "Fundamental definitions, theorems, and proofs". Quite apart from content, you should probably be aware that very few mathematics textbooks at the secondary level (with the exception of Geometry!) are organized along "definitions, theorems, and proofs". Students learn techniques for solving problems, but those techniques are not organized into a logico-deductive structure, in which some properties are taken as primitive and others derived. That is not to say that such a structure does not exist at all, but it is not made explicit; the word "Theorem" is almost never used, and when it is used, normally a Theorem is merely illustrated, not proved.