Chapter 1 Review (Day 6)

The first thing you might be wondering is, what happened to Section 1-9? In this section, the U of Chicago text presents the Triangle Inequality, which seems to be an important concept for students to learn. But here's where the problem lies:

"In this book we treat the Triangle Inequality as a postulate. That is, we do not prove it from other postulates."

The problem is that the Triangle Inequality is actually a theorem. It is, contrary to what's written in the book, provable from other postulates. And so, following David Joyce, we shouldn't label it as a postulate.

Franklin Mason ("Dr. M"), meanwhile, indeed gives a proof of the Triangle Inequality -- and his proof goes all the way back to Euclid. It is Euclid's Proposition 20, where the ancient geometer proves that in triangle ABC, AB + AC > BC:

Euclid, Proposition 20:
"Draw BA through to the point D, and make DA equal to CA. Join DC.
Since DA equals AC, therefore the angle ADC also equals the angle ACD.
Therefore the angle BCD is greater than the angle ADC.
Since DCB is a triangle having the angle BCD greater than the angle BDC, and the side opposite the greater angle is greater, therefore DB is greater than BC.
But DA equals AC, therefore the sum of BA and AC is greater than BC.
Similarly we can prove that the sum of AB and BC is also greater than CA, and the sum of BC and CA is greater than AB.
Therefore in any triangle the sum of any two sides is greater than the remaining one."

We can follow the proof to see where Euclid's reasoning comes from.

-- The first line "Draw BA..." is essentially the Point-Line-Plane Postulate (with DA = CA coming from the Ruler Postulate part of the P-L-P Postulate).
-- The second line "Since DA..." is the Isosceles Triangle Theorem (since triangle ACD is isosceles). This is in Section 5-1 of the U of Chicago text.
-- The third line "Therefore the angle..." comes from the Angle Addition Postulate (Section 3-1), the Equation to Inequality Property, and Substitution Property of Equality.
-- The fourth line "Since DCB..." comes from the Unequal Angles Theorem, which isn't given in the U of Chicago text until Section 13-7 as it depends on indirect proof.

So the U of Chicago text does provide all of the results needed to prove the theorem, but not until we reach Chapter 13. (Here in Chapter 1, we haven't even defined angle yet!) Dr. M includes a two-column proof with eight steps that the students are to fill in. This result is part of his Chapter 5, which gives some of the theorems that U of Chicago provides in 13-7, including the Unequal Angles Theorem (which Dr. M calls the "Triangle Angle Side Inequality").

Now both Dr. M and U of Chicago derive the Unequal Angles Theorem (that is, TASI) from the (Triangle) Exterior Angle Inequality. But then again, the paths diverge. In Dr. M the TEAI is a postulate (but recall that an old blog post of his derives this from SAS) while in U of Chicago, it is derived from the Exterior Angle (Equality) Theorem, which itself goes back to the sum of the angles of a triangle being 180. (Actually, I've seen other high school geometry texts that do derive the Triangle Inequality from the Unequal Angles Theorem but then proceeds back to the Exterior Angle Equality as in U of Chicago.)

But here's the thing -- this is Euclid's Proposition 20, and recall that all of his propositions up to 28 don't require the Parallel Postulate. Thus even though the sum of the angles of the triangle requires Playfair, the Triangle Inequality doesn't require Playfair. (In particular, that an exterior angle of a triangle is the sum of the two remote interior angles requires Playfair, but that the exterior angle is merely greater than the two remote interiors doesn't require Playfair.)

What, then, should I do about the Triangle Inequality? I could follow the Joyce ideal -- never present a statement as a postulate when it can be proved as a theorem, prove every theorem, and never use a statement requiring Playfair to derive a statement that is provable without Playfair. This is what most professional mathematicians do -- it's considered more elegant to have as few postulates and as many proved statements as possible.

But high school students are not professional mathematicians. Student understanding has priority over mathematical elegance. Recall that last month, I distinguished low-, medium-, and high-level proofs -- with high-level proofs being the ones that students are usually asked to complete. Medium-level proofs of theorems are used to prove high-level results, and low-level proofs of theorems are used to prove both medium- and high-level results. Surely the Triangle Inequality is a low- or medium-level result -- students are more likely to use the Inequality than to prove it. Since the U of Chicago text doesn't expect students to prove it, the text declares the Inequality to be a postulate. Dr. M expects students to prove it, with the prove involving the Exterior Angle Inequality. But he doesn't expect students to prove the TEAI, so he declares the TEAI to be a postulate.

Also having priority over mathematical elegance are the Common Core Standards. I wish to focus on proving the results that the standards explicitly ask students to prove. And the standards currently don't mention the Triangle Inequality at all. But, when I performed a Google search just to double check, I discovered that an older version of the standards do mention both the Triangle Inequality and TEAI as results to prove! (Of course, the standards didn't mention whether TEAI should be derived from Playfair or SAS.)

I've decided to delay the Triangle Inequality until it can be proved. I could try to prove it before giving the Playfair Postulate in the first semester (since the Inequality doesn't derive from Playfair), but I've already mentioned that I'll wait until second semester to give indirect proofs -- once again, in order to avoid confusing students with indirect proofs. And so I don't have to decide whether to give the Playfair or the SAS proof of the TEAI until then.

Instead, I now give a quick review of the sections of Chapter 1 that I did cover. This is mostly derived from the Vocabulary and Questions on SPUR Objectives that appear in the text, except that I skip over sections that I didn't cover. Notice that the text gives vocabulary and questions from Sections 1-4 and 1-5, and I did cover these sections, but only as a possible Opening Activity (and teachers may have come up with their own Opening Activity). So my review worksheet only covers 1-6 through 1-8. (If you prefer, you may think of these exercises as Section 1-8 Part 2, since that's where most are from.)