Wednesday, October 8, 2014

The Plane Separation Postulate (Day 44)

As I have already stated, I am basing my postulates and theorems on Wu's website. But now that I have committed to the Wu system, I must stay with it or risk circularity. I already mentioned how in the second semester, I want to do the U of Chicago's Chapter 12 on similarity before Chapter 11 on coordinates, yet the proofs in Chapter 12 depend on theorems from the earlier chapter. To avoid circularity, I must see how Wu proves the theorems in Chapter 12 without using Chapter 11 proofs.

In order to prepare for this, I check to see how Wu proves his theorems. Now Wu's theorems are proved using Wu's axioms (postulates), which he labels as (A1), (A2), and so on. We've already checked his theorems one by one, but let's look at his axioms and compare them to postulates already given in the U of Chicago text:

-- (A1) states, "Through two distinct points passes a unique line." This is clearly the Unique Line Assumption part of the U of Chicago's Point-Line-Plane Postulate.

-- (A2) is the Parallel Postulate of Playfair. The U of Chicago discusses the Parallel Postulate in both Chapters 5 and 13.

-- (A3) Line Separation and (A4) Plane Separation have no counterparts in U of Chicago.

-- (A5) discusses distance, so it corresponds roughly to the Distance Assumption part of the U of Chicago's Point-Line-Plane Postulate.

-- (A6) discusses angle measure, so it corresponds to the Angle Measure Postulate.

-- (A7) discusses the properties of rigid motions. So it corresponds to the Reflection Postulate -- other isometries such as rotations and translations aren't necessary in the U of Chicago because the text defines these in terms of reflections.

So most of these already appear in the U of Chicago in some form. But the postulate that I wish to highlight here is (A4). Wu calls this "Plane Separation":

(A4) (Plane Separation) A line l separates the plane into two non-empty
convex subsets, L and R, called half-planes, so that:

(i) Every point in the plane is in one and only one of the sets L, R, and l.

(ii) If two points A and B in the plane belong to different half-planes, then
the line segment AB must intersect the line l.

Notice that Wu uses the letter L twice -- an italicized L to denote a line, and then a script L to denote the "left half-plane" (so that R is the "right half-place"). I have rewritten the axiom so that a lowercase italicized l denotes a line instead.

In some ways, this postulate is obviously true -- but then again, that's exactly what makes it a "postulate" -- a statement that is obviously true, yet not provably true. Wu admits that (A4) is used to prove boring, tedious theorems such as, if two angles share a side (a ray) and the other sides of the angles lie in the same half-plane bounded by the line containing the same side, then these other sides coincide iff the angles have the same measure. No student should have to prove that theorem, and any proof requiring this theorem (e.g., the proof that two angles are congruent iff they have the same measure) just uses hand-waving to avoid such overly formalist tedium.

But (A4) does have some use in a high school classroom -- especially the Common Core classroom where reflections and rotations rule the day. Wu points out that the converse of part (ii) of that theorem -- namely that if A and B are points not on l and AB intersects l, then A and B belong to different half-planes -- follows immediately from the definition of "convex."

So here is how (A4) relates to reflections:

Let A be any point not on a line l, and let B be its reflection image over l. Then A and B are on opposite sides of l (that is, they belong to different half-planes).

Paragraph Proof:
By the definition of reflection, l is the perpendicular bisector of AB -- so l intersects AB. Therefore, by the Plane Separation Postulate, A and B belong to different half-planes. QED

It seems obvious that if we reflect a figure on one side of a mirror over that mirror, then the image appears on the other side, but this is the actual proof of that statement. And we may extend this to 180-degree rotations:

Let A be any point distinct from another point P, and let B be the rotation image of 180 degrees around P. Then A and B are on opposite sides of any line passing through P that doesn't pass through either A or B.

Suppose l is a line through P but not A or B. Now the line AB may be thought of as the angle APB -- a straight angle of 180 degrees since we have a 180-degree rotation. So line l clearly intersects AB at point P -- so A and B belong to different half-planes. QED

But why is this important? For one thing, in double-checking our proofs, there may be a gap in the proof of the Alternate Angles Test Theorem. In particular, we proved that the rotation image of line AB is line CD -- but not necessarily that the rotation image of ray AB is ray CD. It could be that the image of ray AB is on the same side of the transversal as that ray. But if this were true, it would make the same-side interior angles have the same measure, not the alternate interior angles!

The whole point of alternate interior angles is that the angles are on "alternate" -- that is, opposite -- sides of the transversal. It may appear obvious by a picture that rotating an interior angle by 180 degrees gives only its alternate, not same-side, interior angle. But the axiom that tells us what it even means for two points to be on opposite sides of a line is the Plane Separation Postulate (A4). And the theorem derived from it for rotations proves that the rays (and therefore the angles) are on opposite sides of the transversal. This explains why it's alternate -- and not same-side -- interior angles that have the same measure.

A few more points about Wu's axioms -- (A3) is Line Separation -- just as a line divides a plane into two half-planes, a point divides a line into two half-lines. I suspect that if this appeared in U of Chicago, it would call (A3) a theorem and prove it using coordinates and properties of real numbers (such as Trichotomy), the same way that the text can have a Betweenness Theorem rather than a Segment Addition Postulate.

There's also one more postulate, (A8), the Crossbar Axiom. Like Plane Separation, Crossbar makes a statement that appears obvious but isn't proved. Crossbar actually shows up in a proof of the Isosceles Triangle Theorem -- one of the first theorems of Chapter 5. I don't plan on including Crossbar on my blog -- I might be able to use Plane Separation for Reflections instead, or just try to hand-wave it away as most texts do.

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