Over the years I've had several problems with Lesson 1-9, and here's why. Five years ago -- by which I mean the 2014-15 school year -- I noticed that Lesson 1-9 presents the Triangle Inequality as a postulate, when it's in fact provable using the theorems of Lesson 13-7. And so I decided to delay Lesson 1-9 until after 13-7, so we could prove the Triangle Inequality Theorem.
But then four years ago -- the 2015-2016 school year -- I juggled Chapter 13 around again. I ended up covering other lessons in Chapter 13 at various times, but never 13-7. And because I never posted Lesson 13-7, I'd never post 1-9 either. (Recall that the new Third Edition of the text no longer has our version of Chapter 13.)
This is what I wrote last year about today's lesson:
And that takes us to the topic of today's worksheet -- the Triangle Inequality. In the U of Chicago, the Triangle Inequality was given as a postulate, yet it can be proved as a theorem. Many texts, including the Glencoe text, do prove the Triangle Inequality as a theorem, and this is what we will do.
The proof of the Triangle Inequality begins in Glencoe's Section 5-2, where we must prove two theorems, which the U of Chicago calls the Unequal Sides and Unequal Angles Theorems. My student told me that he had no problem understanding these two theorems -- he wanted just a quick review of Indirect Proof in Section 5-3 before moving on to the Triangle Inequality in 5-4. (This is why I'm squeezing in the Triangle Inequality now, rather than prove only the Unequal Sides and Unequal Angles Theorems today and save the Triangle Inequality for next week.)
Dr. Franklin Mason also proves these theorems. In many ways, Dr. M's Chapter 5 is similar to the same numbered chapter in Glencoe, except that Dr. M saves the concurrency results for a separate chapter, Chapter 10. Both Dr. M and Glencoe follow the same sequence of theorems, in which each theorem is built from the previous theorem in the list:
Exterior Angle Theorem (abbreviated TEAE in Dr. M)
Exterior Angle Inequality (TEAI)
Unequal Sides Theorem (TSAI)
Unequal Angles Theorem (TASI)
Triangle Inequality (too important to be abbreviated!)
SAS Inequality (Hinge)
The U of Chicago follows the same pattern, except that the Unequal Angles Theorem is not used to prove the Triangle Inequality. Instead, the Triangle Inequality is merely a postulate. And since Unequal Angles isn't used to prove the Triangle Inequality, the U of Chicago didn't have to wait until
Chapter 13 to state the Triangle Inequality. Instead, the Triangle Inequality Postulate is given in Chapter 1, and the SAS Inequality, which depends on that postulate in its proof, is given in Chapter 7, still well before Chapter 13.
My blog attempted to restore the Dr. M-Glencoe order by delaying the Triangle Inequality. But I screwed up by not delaying the SAS Inequality as well. This is why I plan on delaying SAS Inequality, so that the full logical sequence is given. Of course all I did that year was make things worse!
But the first four theorems in the list are proved in U of Chicago's Section 13-7. Since I briefly mentioned the Exterior Angle Theorem (TEAE) at the end of the first semester, and the TEAI follows almost trivially from TEAE, my worksheet skips directly to the Unequal Sides Theorem. Its proof is given in the two-column format. Here I reproduce that proof, starting with a Given step:
Unequal Sides Theorem (Triangle Side-Angle Inequality, TSAI):
If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side.
Given: Triangle ABC with BA > BC
Prove: angle C > angle A
Proof:
Statements Reasons
1. Triangle ABC with BA > BC 1. Given
2. Identify point C' on ray BA 2. On a ray, there is exactly one point at a given distance from
with BC' = BC an endpoint.
3. angle 1 = angle 2 3. Isosceles Triangle Theorem
4. angle 2 > angle A 4. Exterior Angle Inequality (with triangle CC'A)
5. angle 1 > angle A 5. Substitution (step 3 into step 4)
6. angle 1 + angle 3 = angle BCA 6. Angle Addition Postulate
7. angle BCA > angle 1 7. Equation to Inequality Property
8. angle BCA > angle A 8. Transitive Property of Inequality (steps 5 and 7)
The next theorem is proved only informally in the U of Chicago. The informal discussion leads to an indirect proof.
Unequal Angles Theorem (Triangle Angle-Side Inequality, TASI):
If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.
Indirect Proof:
The contrapositive of the Isosceles Triangle Theorem is: If two angles in a triangle are not congruent, then sides opposite them are not congruent. But which side is opposite the larger angle? Because of the Unequal Sides Theorem, the larger side cannot be opposite the smaller angle. All possibilities but one have been ruled out. The larger side must be opposite the larger angle. QED
My student told me that he wanted to see one more indirect proof before showing him the Triangle Inequality, so why not show him this one? The initial assumption is, assume that the longer side is not opposite the larger angle. Since the angle opposite the longer side is not greater than the angle opposite the shorter side, the former must be less than or equal to the latter. And these are the two cases that lead to contradictions of Isosceles Triangle Contrapositive and Unequal Sides as listed in the above paragraph proof.
Now finally we can prove the big one, the Triangle Inequality. This proof comes from Dr. M -- but Dr. M writes that his proof goes all the way back to Euclid. Here is the proof from Euclid, where he gives it as his Proposition I.20:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI20.html
Here is the two-column proof as given by Dr. M. His proof has eight steps, but I decided to add two more steps near the beginning. Step 1 is the Given, and Step 2 involves extending a line segment, so that it's similar to Step 2 of the Unequal Sides proof. Indeed, the proofs of Unequal Sides and the Triangle Inequality are similar in several aspects:
Triangle Inequality Theorem:
The sum of the lengths of two sides of any triangle is greater than the length of the third side.
Given: Triangle ABC
Prove: AC + BC > AB
Proof:
Statements Reasons
1. Triangle ABC 1. Given
2. Identify point D on ray BC 2. On a ray, there is exactly one point at a given distance from
with CD = AC an endpoint.
3. angle CAD = angle CDA 3. Isosceles Triangle Theorem
4. angle BAD = BAC + CAD 4. Angle Addition Postulate
5. angle BAD > angle CAD 5. Equation to Inequality Property
6. angle BAD > angle CDA 6. Substitution (step 3 into step 5)
7. BD > AB 7. Unequal Angles Theorem
8. BD = BC + CD 8. Betweenness Theorem (Segment Addition)
9. BD = BC + AC 9. Substitution (step 2 into step 8)
10. BC + AC > AB 10. Substitution (step 9 into step 7)
To help my student out, I also included another indirect proof in the exercises. We are given a triangle with two sides of lengths 9 cm and 20 cm, and we are asked whether the 9 cm side must be the shortest side. So we assume that it isn't the shortest side -- that is, that the third side must be even shorter than 9 cm. This would mean that the sum of the two shortest sides must be less than 9 + 9, or 18 cm, and so by the Triangle Inequality, the longest side must be shorter than 18 cm. But this contradicts the fact that it is 20 cm longer. Therefore the shortest side must be the 9 cm side. QED
Notice that the U of Chicago text probably expects an informal reason from the students. A full indirect proof can't be given because this question comes from Section 1-9, while indirect proofs aren't given until Chapter 13.
Returning to Lesson 1-9, let's post the worksheets. First of all, since I'm now following the U of Chicago order, students are no longer responsible for a proof of the Triangle Inequality, so I only post the questions that don't depend on a proof.
On the other side, I post a review for the Chapter 1 Test. Recall that the Chapter 1 Test must be given on Day 20, or tomorrow, since Day 21 is Lesson 2-1. If there are eight or fewer lessons in a chapter, then there's a separate review day, but if there are nine lessons in a chapter, then the ninth lesson falls the day before the test.
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