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.

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.

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 l_{2} 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.

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 l_{1} and l intersect at right angles, then
l_{1} 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 l_{1} and
l_{2} perpendicularly. Then l_{1} and l_{2} are parallel.

(Compare this to Theorem 1.4.1, which said that if l_{1} and l_{2}
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 l_{1} and l_{2} 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 l_{1} and l_{2} will have to bend a little to meet at R.)

By Axiom 8, there exists an isometry that is a reflection through l. Because l_{1}
and l_{2} are perpendicular to l, this
isometry must "flip" line l_{1} onto itself and also map l_{2} 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 l_{1} and l_{2} intersect
at two points -- R and R'. But axiom 1 says that the line joining R and R' is unique, so then
l_{1} and l_{2} would have to be the same line.

If the reflected image of R is R itself, then lines l_{1} and l_{2} intersect
at point R on line l. If l_{1} and l_{2} 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 l_{1} and l_{2} 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 l_{1} and l_{2} 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.//