MATH325: Lecture 3


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Review of Isometries, SSS

Exercise 1.9: Describe an isometry of the plane which has no fixed points and is not a translation.

Take a piece of paper and flip and slide it around for a while. What do you think? (Discussion.)

Do you feel like you understood the proof of "side side side"? If you were asked to prove it on an exam, could you do it? Reading a proof is harder than reading your email -- don't hesitate to "follow along" with paper and pencil if you can't visualize what's going on in the proof.

You might find it interesting to compare the version of the proof given in lecture to the one in the book. They're essentially the same proof, but the one in the book is more precise. It's normal to understand a proof somewhat "casually" at first, then go back over it more formally. When writing proofs for homework, you should expect to produce at least one draft of the proof before writing up the final verison.

Homework: Prove Exercise 1.15 and answer one of the questions on the message board.

Axiomatic Systems, Reprise

Recall that our goal is to prove all the theorems we know to be true, starting from a small number of axioms, definitions and assumptions. So far we've managed to avoid using Axiom 5 in any of our proofs (it's standard to avoid Axiom 5 for as long as possible -- even Euclid did it.) However, we've reached a point where we can't prove the things we want to prove without using Axiom 5.

Before we go forward, we'll prove that our Axiom 5 is essentially the same as Euclid's fifth postulate. Before we do that, we'll take a look at what definitions, lemmas and assumptions the text book uses to prove the theorems in section 1.4.

1.4: Parallel Lines

Theorem 1.4.1 says that if you have two parallel lines and you draw a line perpendicular to one of them, it's perpendicular to the other. To prove this the book uses Definition 4 (to prove that certain angles are right angles), Lemma 1.4.2 (to construct the point of intersection of the perpendicular line with the original line) and Axiom 5 (a broad outline of the proof would be: "if Theorem 1.4.1 is not true, then Axiom 5 is not true. We have declared Axiom 5 to be true, so Theorem 1.4.1 must also be true.")

Theorem 1.4.1 is an example of a theorem that we cannot prove without using Axiom 5. Theorems like 1.4.1 that tell you about the angles created when a transversal line crosses a pair of parallel lines are common and useful, so we want to add Axiom 5 to our collection of assumptions.

A theorem is a mathematically proven result.

A lemma is a mathematically proven result that is used to prove a theorem.

A corollary is a result that is easy to prove once you've proven the theorem it is a corollary to.

Corollary 1.4.5 is a proof of Euclid's fifth postulate. This proof uses our Axiom 5. (Naturally -- if it were possible to prove the fifth postulate without using Axiom 5, a lot of mathematicians' jobs would have been a lot easier and the universe would be a less interesting place.) It also uses Axioms 6 and 7, Definition 4, Corollary 1.4.4 and an "obvious" fact about triangles that is proved in Chapter 9.

How does Corollary 1.4.5 use Theorem 1.4.1? Corollary 1.4.5 is based on Corollary 1.4.4, which uses Theorem 1.4.3 (I'm pretty sure the bit where it says it's using 1.4.1 is a typo). Theorem 1.4.3 is the converse of Theorem 1.4.1, which may or may not be used in proving Corollary 1.4.5.

Now that we've reviewed section 1.4 and thought some more about the foundations proofs are built on, let's get back to the job of proving that our Axiom 5 (part of the foundation of the reasoning in this text) is equivalent to the fifth postulate (part of the foundation of the "classical geometry" that appears in almost all geometry text books.) This proof is necessary because different foundations can lead to different conclusions -- if the conclusions in this textbook don't match the conclusions in the book you're teaching from, this book is less useful to you than it might be.

Axiom 5: Given a line l and a point P not on l, there exists a unique line l2 through P which does not intersect l.

Fifth Postulate: 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.

(In other words: "Suppose a line meets two other line so that the sum of the angles on one side is less than two right angles. Then the other two lines meet at a point on that side.")

If we can show that our fifth axiom is somehow equivalent to Euclid's fifth postulate, anything that was true for Euclid will be true in our system too. To prove this equivalence, we use our axiom 5 to prove Euclid's fifth postulate (to see that when we assume axiom 5, Euclid's fifth postulate must also be true) then use Euclid's fifth postulate to prove our axiom 5 (so assuming the fifth postulate means axiom 5 must be true). Since each assumption can be used to prove the other, neither is "more powerful"; assuming one yields the same results as assuming the other.

The text uses axiom 5 to prove the fifth postulate; the proof of axiom 5 based on the fifth postulate is Exercise 1.21.

Theorem 1.4.3

Remember that the result we want (Corollary 1.4.5) depends on a lot of earlier results. Theorem 1.4.3 is the theorem that 1.4.5 is the corollary to.

Let's assume that the sum of the measures of supplementary angles is 180°. This seems reasonable based on Definition 4, which defines right angles as a pair of congruent supplementary angles.

We also assume the converse of Exercise 1.17, i.e. that if l1 and l intersect at right angles, then l1 is congruent to itself by a reflection through l. (Extra credit if you can prove this.)

Next we prove Theorem 1.4.3; the result we want is a corollary to this theorem.

Theorem 1.4.3: Suppose l intersects two distinct lines l1 and l2 perpendicularly. Then l1 and l2 are parallel.

(Compare this to Theorem 1.4.1, which said that if l1 and l2 are parallel, then a line perpendicular to one must be perpendicular to both.)

Discussion of Proof: The proof proceeds by contradiction. In other words, the author starts out by saying "what if the lines aren't parallel?" He shows that if the two lines aren't parallel to each other then they must be the same line; but the theorem says the two lines are distinct, so they can't be the same line, so it can't be true that they're not parallel.

Proof: What if the lines l1 and l2 aren't parallel? Then they must intersect each other at some point; call it R. (Sketch the three lines and the point on your paper -- the lines l1 and l2 will have to bend a little to meet at R.)

By Axiom 8, there exists an isometry that is a reflection through l. Because l1 and l2 are perpendicular to l, this isometry must "flip" line l1 onto itself and also map l2 onto itself. But what happens when you reflect R through l? Either you get a new point R' or R is on top of l.

If the reflected image of R is a new point R', then l1 and l2 intersect at two points -- R and R'. But axiom 1 says that the line joining R and R' is unique, so then l1 and l2 would have to be the same line.

If the reflected image of R is R itself, then lines l1 and l2 intersect at point R on line l. If l1 and l2 are distinct there must be an angle between them, but then we could find a pair of supplementary angles whose measures added to more than 180°. Therefore, since l1 and l2 are both perpendicular to l and both intersect l at the same point R, they must both be the same line. Whether R is on line l or not, we've concluded that if l1 and l2 are both perpendicular to l and are not parallel then they must be the same line. Therefore the only way for them to be distinct lines perpendicular to l is for them to both be parallel to l.//