Today, we will use Wu's First Theorem to derive the Parallel Tests -- that is, the statements that if two lines are cut by a transversal in a way to make alternate interior (or corresponding) angles congruent, then the lines are parallel.
But there are a few things to point out about this lesson. First of all, let's remind everyone where the name "Parallel Tests" comes from -- Dr. Franklin Mason. Dr. M then uses the name "Parallel Consequences" to refer to the statements of the form "if two lines are parallel, then..." -- in other words, they are the converses of the Parallel Tests.
But why should we distinguish between Parallel Tests and Parallel Consequences? For one thing, in geometry, a statement can be true even while its converse is false. "If a quadrilateral is a rhombus, then it is a parallelogram" is one such statement. But there are so many true statements with true converses -- the Parallel Tests among them -- that students fail to make the distinction. This is especially true when looking at the exercises typical in many geometry texts. An exercise based on a Parallel Consequence may give a diagram of two lines cut by a transversal, with one angle labeled with a degree value and another labeled with a variable, with the instruction:
If l | | m, then find the value of x.
And then, an exercise for a Parallel Test would be the same diagram, with the instruction:
Find the value of x that would make l | | m.
So there's no surprise that the students confuse a statement with its converse. Moreover, it turns out that in non-Euclidean hyperbolic geometry, the Parallel Tests are true, yet their converses are false (just like "if a quadrilateral is a rhombus, then it is a parallelogram")! So the Parallel Consequences all require a Parallel Postulate to prove, while the Parallel Tests do not.
In Dr. M's original formation of geometry, he uses triangle congruence to prove the Triangle Exterior Angle Inequality, then in turn uses the TEAI to prove the Alternate Interior Angles Test, and finally uses AIA to prove the other tests. But as I mentioned earlier, Dr. M eventually dropped the proof of TEAI and just included it as a postulate instead. Now I notice that he's taken this one step further, and dropped TEAI from Chapter 4 on parallels entirely. Now Dr. M simply has a Corresponding Angles Postulate, more in line with traditional texts. Notice that both Dr. M and the U of Chicago, in its Section 3-4, use the name "Corresponding Angles Postulate" to refer to that Parallel Test -- other texts use that name to refer to the Parallel Consequence. Once again, whenever Dr. M, or any other author, drops the proof of a theorem and replaces it with a postulate, the implication is that the proof would be too confusing for students to understand, which is what really matters.
But although I'm using Dr. M's Tests/Consequences distinction, our proofs won't follow Dr. M, but rather Wu. Now Wu never uses TEAI to prove any Parallel Test. Instead, he uses 180-degree rotations in his proof. And so we, unlike Dr. M, will still prove the Alternate Interior Angles Test, and derive the Corresponding Angles Test from it.
Notice the primacy of AIA here. Most texts assume the Corresponding Angles Test as a postulate and then prove the AIA test from Corresponding Angles. Dr. M does this now, as well as the U of Chicago in its Section 5-6. But Dr. M originally derived AIA from TEAI. But it actually is possible to derive the Corresponding Angles Test, rather than AIA, from TEAI. Believe it or not, I have a textbook that actually does this -- after all, the angle corresponding to an interior angle is also exterior to the triangle formed by two intersecting lines and a transversal (so that there's not much difference between an AIA proof and a Corresponding Angles proof derived from TEAI).
But Wu's proof always leads directly to Alternate Interior Angles. In Wu, it is Theorem 12:
"Thoerem [sic] 12. If a pair of alternate interior angles [...]
of a transversal with respect to two lines are equal, then the lines are parallel."
(Actually, Wu does mention "corresponding angles" here, but makes it clear that the proof with corresponding angles is just a corollary to the main proof with AIA.)
You may notice that yesterday we proved Theorem 1, and now we prove Theorem 12. What happened to Wu's Theorems 2 through 11? Let's see what these theorems are:
Theorem 2: Two Perpendiculars Theorem (U of Chicago Section 3-5, already proved on this blog)
Theorem 3: Uniqueness of Perpendiculars Theorem (already proved on this blog using reflections, but Wu goes the other way and uses this theorem to prove that reflections exist)
Theorem 4: Properties of a Parallelogram Theorem. (U of Chicago Section 7-6). Actually, I like Wu's proof here and may include it when we get to parallelograms in Chapters 5, 6, and 7.
Theorems 5 and 6: SAS and ASA (U of Chicago 7-2, will include when we reach Chapter 7)
Theorem 7: The Isosceles Triangle Theorem (Section 5-1, will include when we reach Chapter 5)
Theorems 8 and 9: HL and SSS (U of Chicago Chapter 7)
Theorems 10 and 11: The Parallel Consequences and the Triangle-Sum Theorem
And we'll get to the Parallel Consequences when we discuss the Parallel Postulate in Chapter 5.
It must be pointed out that the coverage of parallel lines in U of Chicago is rather segmented. We discuss corresponding angles in Section 3-4, yet alternate interior angles are postponed all the way until Section 5-6 -- when they are used to prove "if a quadrilateral is a rhombus, then it is a parallelogram" as a theorem. And same-side interior angles don't appear at all -- but then again, in Section 5-5, the Trapezoid Angle Theorem tells us that consecutive angles of a trapezoid are supplementary -- and these consecutive angles are actually just same-side interior angles. (Notice that Dr. M calls them "consecutive interior angles.") But the text never discusses same-side (consecutive) interior angles where there is no trapezoid. It's as if for the U of Chicago, the Parallel Tests and Consequences don't matter unless they are used to prove properties of quadrilaterals.
I've decided not to do this. I'm just going to give all three Parallel Tests -- Alternate Interior Angles, Corresponding Angles, and Same-Side Interior Angles -- right here, right now. So I'm jumping from Section 3-4 to 5-6 (with a little bit of 5-5 thrown in).
Here is Wu's proof of Theorem 12, the Alternate Interior Angles Test. Here Wu uses so many confusing subscripts that I've decided to change them. The two lines to be proved parallel will be called m and n, with the transversal still called l, which intersects m at point A and n at B. The points C and D are the same as Wu's, but without the subscripts.
Given: Angles CAB and ABD have the same measure.
Prove: m | | n
Let O be the midpoint of AB, and rotate 180 degrees around O. Since O is on l, the rotation image of l is exactly l. Now where is A"? A" must lie on l and is the same distance from O as A is. This point must be exactly B (Ruler Postulate). Now where is C"? We look at angle CAB and notice that its image, ABC", must have vertex B, have the same measure as CAB, and lie on the alternate side of the transversal as CAB. This angle must be exactly ABD. (Notice that we can't conclude that C" is exactly D -- only that BC" and BD are the same ray.) Now where is m"? We see that m" contains points C and D, so m" is exactly n. By Theorem 1 from yesterday, the 180-degree rotation image of any line is parallel to the line. Therefore m || n. QED
Of course, we don't need a proof to see the truth of this theorem. Just take a sheet of paper and draw two lines cut by a transversal with equal alternate interior angles, and turn it upside-down. The image looks identical to the original right-side-up picture.
And so we restore the primacy of Alternate Interior Angles Test. This primacy goes all the way back to Euclid, where he gives this as Proposition 27:
"If a straight line falling on two straight lines makes the alternate [interior] angles equal to one another, then the straight lines are parallel to one another."
Euclid uses Proposition 16 -- which is essentially the Triangle Exterior Angle Inequality (just as Dr. M originally used) -- to prove Proposition 27. Proposition 28 gives the Corresponding and Same-Side Interior Angle Tests, and Proposition 29 gives the Parallel Consequences. Notice that Euclid needs his Fifth Postulate to prove Proposition 29, but not 27 or 28.
Notice that the Two Perpendiculars Theorem mentioned earlier is also a Parallel Test -- Dr. M calls it the Perpendicular Transversal Test. Indeed, all of our tests can be traced back to either of our first two Parallel Tests -- the Two Perpendiculars Theorem and the Line Parallel to Mirror Theorem.
Today's lesson corresponds to Lesson 18 from New York State. My exercises contain a combination of Questions from 3-4 and 5-6 in the text.