This week, I'm returning to Chapter 3 of the U of Chicago text, to pick up some of the lessons we skipped. But even then, I'm not simply doing the second half of Chapter 3 in order. First, I'm still bypassing Section 3-4, since I want to save the Corresponding Angles Postulate until after I'm ready to show Hung-Hsi Wu's method for deriving this property -- which I plan on doing soon.
Officially, I'm doing Section 3-5 now, but then again, not really. Let's consider the contents of this particular section:
-- The definition of perpendicular has already been covered. I'm moved it to Section 3-2 when I defined right angles, because I wanted to get it in before jumping to Chapter 4 on reflections, since reflections are defined in terms of the perpendicular bisector.
-- The Perpendicular to Parallels Theorem is an interesting case. Last week, I mentioned that there are certain interesting and important theorems that are derived from Perpendicular to Parallels -- and these include the properties of translations as well as some of the concurrency theorems. I said that it might be better just to include this as a postulate and use it to prove those other theorems. And as I think about it more and more, I like the idea of including this as a postulate, then using it to prove those other theorems as well as Playfair's Parallel Postulate, and then finally using Playfair to prove the other Parallel Consequences following Dr. Franklin Mason.
Now as I imply in this post, most of what I write on this blog is derived from mathematicians like Dr. M and Dr. Wu, who have written extensively about Common Core Geometry. But my plan to include Perpendicular to Parallels as a postulate appears to be original to me. I've searched and I have yet to see any text or website who will derive all the results of parallel lines from a Perpendicular to Parallels Postulate. Then again, what I'm doing here is, in some ways, as old as Euclid.
Let's look at Playfair's Parallel Postulate again:
Through a point not on a line, there is at most one line parallel to the given line.
This is a straightforward, easy to understand rendering of a Parallel Postulate, and Dr. M uses this postulate to derive his Parallel Consequences. But let's look at Euclid's Fifth Postulate, as written on David Joyce's website:
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
No modern geometry text would word its Parallel Postulate in this manner. For one thing, even though the use of degrees to measure angles dates back to the ancient Babylonians, Euclid never uses degrees in his Elements. So the phrase "less than two right angles" is really just Euclid's way of writing "less than 180 degrees." Indeed, in Section 13-6, the U of Chicago text phrases Euclid's Fifth Postulate as:
If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180, then the lines will intersect on that side of the transversal.
But let's go back to the Perpendicular to Parallels Theorem as stated in Section 3-5:
In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Now count the number of right angles mentioned in this theorem. The transversal being perpendicular to the first line gives us our first right angle, and the conclusion that the transversal is perpendicular to the second line gives us our second right angle. So we have two right angles -- just like Euclid! So in some ways, making the Perpendicular to Parallels Theorem into our Parallel Postulate is making our postulate more like Euclid's Fifth Postulate, not less.
Of course, if we wanted to make our postulate even more like Euclid's, we could write:
If a plane, if a transversal is perpendicular to one line and form an acute angle (that is, less than right) with another, then those two lines intersect on the same side of the transversal as the acute angle.
But this would set us up for many indirect proofs, which I want to avoid. So our Perpendicular to Parallels Postulate is the closest we can get to Euclid without confusing students with indirect proofs.
So this is exactly what I plan on doing. Since we're in Section 3-5, the section that has Perpendicular to Parallels given, I could include it here -- we don't need to worry about how to prove it since I want to make it a postulate. But I already said that I want to wait until Chapter 5 before including any sort of Parallel Postulate. And so the new postulate will be given in that chapter.
Returning to Lesson 3-5:
-- The Perpendicular Lines and Slopes Theorem must also wait. Common Core Geometry gives an interesting way to prove this theorem, but the proof depends on similar triangles, which I don't plan on covering until second semester.
So that leaves us with only one result to be covered in 3-5: the Two Perpendiculars Theorem:
If two coplanar lines l and m are each perpendicular to the same line, then they are perpendicular to each other.
This theorem doesn't require any Parallel Postulate to prove. Indeed, even though I just wrote that I don't want to use indirect proof, in some ways this theorem is just begging for an indirect proof:
Assume that lines l and m are both perpendicular to line n, yet aren't parallel. Then the lines must intersect (as they can't be skew, since we said "coplanar") at some point P. So l and m are two lines passing through point P perpendicular to n. But the Uniqueness of Perpendiculars Theorem (stated on this about two weeks ago) states that there is only one line passing through point P perpendicular to n, a blatant contradiction. Therefore l and m must be parallel. QED
But this would be a very light lesson indeed if all I included is this one theorem. Because of this, I decided to include a theorem that I mentioned back in July -- the Line Parallel to Mirror Theorem (companion to the Line Perpendicular to Mirror Theorem mentioned a few weeks ago):
Line Parallel to Mirror Theorem:
If a line l is reflected over a parallel line m, then l is parallel to its image l'.
As I mentioned in July, I proved this theorem in the same way that Wu proves for a similar theorem involving 180-degree rotations. In some ways, both are special cases of the following theorem:
Suppose T is a transformation with the following properties:
-- The image of a line is a line.
-- Through every point P in the plane, there exists a line L passing through P such that L is invariant with respect to T -- that is, T maps L to itself.
Then any line that doesn't contain a fixed point of T must be parallel to its image.
Notice that there are three confusing concepts in this theorem -- fixed point, invariant line, and reflection-symmetric figure. On the surface, all three mean the same thing: image of the point, line, or figure is itself. But there is a key difference -- an invariant line simply means that the image of any point on the line is also somewhere on the line -- but it need not be that point itself. In other words, not every point on an invariant line is a fixed point. Similarly, not every point on a symmetric figure needs to be a fixed point, nor every line on a symmetric figure an invariant line.
Now all Common Core transformations satisfy the first property -- that the image of a line is itself some line. (Forget about that Geogebra "circle reflection" where the image of a line can be a circle, since that's not a Common Core transformation.) As it turns out, there are five types of Common Core transformations that satisfy the second property:
-- Any reflection
-- Any translation
-- Any glide reflection
-- Any dilation
-- A rotation of 180 degrees
So notice that the only Common Core transformations not satisfying the second property are rotations of angles other than 180 degrees.
Here is a proof of the lemma. (By the way, "lemma" means a short theorem that is mainly used to prove another theorem.) Let l be the original line and l' its image, and let P be any point on l. Since l doesn't contain any fixed points, the image of P can't be P itself -- so instead, it must be some point distinct from P, which we'll call P'. So of course P' lies on l'. Now the point P lies on some invariant line L -- and by invariant, we mean that P' lies on it. Now through the two points P and P', there is exactly one line, and that line is L, not l. Since P lies on l, it means that P' can't lie on l. But this is true for every point P on l. For every point P on l, P' is not on l. So l can't intersect its image l', since every point on l fails to have an image on l. In other words, l and its image l' are parallel. QED
And so to prove the Line Parallel to Mirror Theorem, it suffices to show that any reflection satisfies the hypotheses of the lemma. We know that the reflection image of a line is a line (part of the Reflection Postulate), and we know that through any point P not on the mirror m, there is a line through P perpendicular to m (Uniqueness of Perpendiculars Theorem) -- which matters because the lines perpendicular to the mirror are invariant lines (Line Perpendicular to Mirror Theorem). So the hypotheses of the lemma are satisfied. Any line parallel to the mirror is parallel to its image. QED
This is a valid proof, but it's inappropriate for a high school geometry class. Providing a lemma that works for a wide variety of transformations and then showing that reflections are the correct type of transformation requires a level of abstraction that we don't expect high school students to have. So instead, we must prove the theorem for each of the types of transformations. But doing it in the case of reflections will prepare the students for the proof from Wu, who does it with 180-degree rotations.
But now we ask, is this a good time to provide the Line Parallel to Mirror Theorem? It makes sense that I'm giving it before the Wu proofs, but does it make sense to put it in the same lesson as the Two Perpendiculars Theorem -- especially since neither theorem is used to prove the other? In some ways, these two theorems do have something in common. These two theorems are our first Parallel Tests -- that is, they are used to prove that lines are parallel. They are both of the form "if two lines have some property (such as each being perpendicular to a third line, or one line being the mirror image of another over a parallel third line), then the lines are parallel."
For these proofs, I use the U of Chicago definition of parallel -- that is, two coplanar lines are parallel if and only if they have no points in common, or they are identical. As I mentioned in July, this U of Chicago definition of parallel allows me to avoid indirect proofs. Notice that the following definitions of parallel are equivalent to "Two lines have no points in common or are identical":
-- If two lines have one point in common, then they have every point in common.
-- If two lines have one point not in common, then they have no point in common.
These last two are written in if-then form, and so are convenient to write as the hypothesis (or Given) and conclusion parts of a two-column proof. The first is used in our proof of the Two Perpendiculars Theorem -- we show that if two lines k and l, both perpendicular to line m, have one point P in common, then they have every point in common. The second is used in our proof of the Line Parallel to Mirror Theorem -- if there is a point P that lies on l but not on m, then the two lines have no point in common.
These proofs will still likely confuse students. So I begin with a reminder of what it means for two lines to be parallel, using our inclusive definition. Usually, I try to include the Questions from the U of Chicago text, but I changed Lesson 3-5 so radically that I could only include a few of them.