## Monday, October 23, 2017

### Lesson 4-7: Reflection-Symmetric Figures (Day 47)

This is what Theoni Pappas writes on page 296 of her Magic of Mathematics:

"Benjamin Franklin was a magic square enthusiast. In fact, while he was a clerk at the Pennsylvania Assembly, he often relieved the tedium of his work by making magic circles."

This page in Pappas is titled "Benjamin Franklin's Magic Circle." Here's a link to Ben Franklin's magic circle, but this link mentions other Franklin magic diagrams, including the order-16 square that we already saw last week.

http://recmath.org/Magic%20Squares/franklin.htm#circle

According to the link, here are all the combinations that add up to 360 or 180:

• The circle uses the integers from 12 to 75 plus another 12 in the center which is used for all summations.
• The eight numbers in each radius plus the central 12 sum to 360.
• The eight numbers in each circle plus the central 12 sum to 360.
• The eight numbers in each eccentric circle plus the central 12 sum to 360. Franklin’s circle apparently shows 20 of these eccentric circles, but is very hard to read. I show only 8 such circles but obviously there are many more!
• Any half circle in the top or bottom, plus half of the center number sum to 180.
• Any half radius plus half of the center number sum to 180.
• Any four adjacent numbers in an ‘almost square’ plus half of the center number sum to 180. For example, 73 + 14 + 15 + 72 + 6 = 180.
• What other combinations are there?
Come to think of it, it's fitting that the magic constant of a circle would be 360 -- as in 360 degrees.

Chapter 7 of Paul Hoffman's The Man Who Loved Only Numbers is called "Survivors' Party." As usual, we begin with a quote -- this time it's a letter:

Dear Ron[ald Graham],

Please send me Uncle Paul's birthdate if you have it. Then we will have something to look forward to and not be so depressed.

Thanks, Ed

P.S. I hope he is giving the SF all sorts of trouble.

And Ronald Graham replies that Erdos -- whose birthday is March 26th, 1913 -- is too busy reading the Book to give anyone much trouble. (By the way, Hoffman never explains who "Ed" is.)

As the title and letter imply, this chapter takes place after the mathematician's death. Every year around his birthday, there's a math conference in Memphis, Tennessee. In 1997, the first birthday after his passing, there is a "survivors' party" held in his honor.

Mathematicians at this reminisce about the days they've spent with Erdos. One of them, John Selfridge, talks about how Anyuka -- Paul's mother -- supports her son as he and Selfridge write a joint paper proving that the product of consecutive integers is never a power. At his funeral six months earlier, Selfridge helps deliver the mathematician's ashes to his mother's grave, and John speaks only the word "Anyuka" at the grave.

Two other mathematicians laugh as they remember how Erdos tells them "Good morning!" or even "Merry Christmas!" and then immediately starts talking about math.

At the University of Memphis where this party takes place, there are several cartoons posted on the walls of the math department. One of them is Doonesbury. The math professor is being rebuked by the chairman of the department because he dared to give a student a B+. The chairman tells him that this is a "new generation of students" who insist on a certain comfort level, and the professor can't fail the students even if they think that 1 + 1 = 3. Even though this comic is dated 1994, I can easily see many traditionalists, most notably Bill, agreeing that this definitely describes the current generation of college students.

That's right -- you may notice that today's post is labeled "traditionalists." That's because there's something I want to do on this special day.

For you see, today, October 23rd, is Mole Day. It's named after the mole as used in Chemistry. Also known as Avogadro's Number, one mole is 6.02 * 10^23. Thus Mole Day is the chemists' Pi Day -- the expression 10^23 is converted into the date 10/23, or October 23? (By the way, an upcoming redefinition of SI will define Avogadro's Number to be exactly 6.022140857 * 10^23.)

The mantissa 6.02 is also used on this special day. We'd like to make it into a time, 6:02, just as the next three digits of pi, 159, are used to denote the time 1:59 on Pi Day. But unfortunately, 6:02 AM is before school while 6:02 PM is after school. Mole Day participants often declare that the holiday is to be celebrated for 12 hours, from 6:02 AM until 6:02 PM. Then this covers the whole school day. Oh, and notice that after I corrected my post count last week, today is my 602nd post!

Here's a link to the official Mole Day website. Apparently this year's theme is "The MOLEvengers."

https://www.moleday.org/

Oh, and the following link gives some Mole Day jokes:

https://www.thoughtco.com/funny-mole-day-jokes-and-humor-608525

Q: How would you describe a stinky chemist?
A: Mole-odorous

Q: What kind of fruit did Avogadro eat in the summer?
A: Watermolens

Q: What kind of test do chemistry students like best?
A: Mole-tiple choice.
Q: Why is Avogadro so rich?
A: He's a multi-mole-ionare!

Today I wish to link to the aforementioned traditionalist Bill. This post is actually dated August 8th, but I actually linked to another article in my August 10th traditionalists post and forgot about Bill. You'll find out soon why I chose Mole Day to return to Bill.

This link is at the Joanne Jacobs website -- one of Bill's favorite places to comment:

http://www.joannejacobs.com/2017/08/hiding-remediation/

As usual, Jacobs herself links to another site, EdSource. I've referenced EdSource directly in the past, but I want to remain at Jacobs since that's where Bill's comments are. The article refers to a remedial math class taken at Cal State Dominguez Hills. (Hey -- that's the second mention this month of the school where I earned my credential!)

For example, a traditional for-credit college algebra class usually meets three times a week for 50 minutes. In contrast, the co-requisite (Aida) Tseggai attended met for an hour and ten minutes three times a week for instructor Cassondra Lochard’s lectures; in addition, students had an extra group hour weekly with a teaching assistant plus one-on-one tutoring.  In contrast to regular classroom protocol, the teaching assistant circulated among the desks during lectures, softly giving advice and reviewing students’ calculations and algebra formulations.

The article goes on to mention Raquel Herrera, a biochemistry student. (Chemistry -- oh, so that's why I'm posting this on Mole Day!) Herrera took -- and failed -- the remedial math class mentioned in the article. And here's Bill's response:

Perhaps Ms. Herrera has been lied to all of her life in school by being given grades rather than actually earning them. I would hope a major in biochemistry would require more math than simply algebra, how would she make it through two semesters of general chemistry, organic chemistry (which was a 2nd year course for chem majors), and Bio-Chemistry?

Another commenter, "Midwestern girl," confirmed that Calculus is indeed required. Bill continues:

That’s what I thought since doing chemistry usually requires math through at least Calc I/II, though my old 9th grade algebra teacher once said “You guys and gals have no trouble with algebra, you just can’t add, subtract, multiply, and divide)…LOLOL
If Ms. Herrera had failed Algebra twice (which she should have taken in high school), she would have never made it through calculus I/II…I can still do basic limits and derivatives, even though I haven’t taken calculus in 35 years (lord knows I have tried to forget it, but that math has warped my fragile little mind – E Cartman)

Let me try to figure out what is going on here. I want to guess Herrera's story here -- why she chose biochem as a major, as well as why she wants to change to "liberal studies" to be a teacher. But I don't wish to use the name Raquel Herrera in this story because my guesses may be false. So instead we will make up a hypothetical student -- let's call him Amadeo (as in Amadeo Avogadro, for whom Avogadro's Number was named). Oh, and the gender change is intentional, since I obviously don't want to give the impression that girls are the only ones who fail remedial math.

Little Amadeo always enjoyed science. When he grows up, Amadeo wants to become a scientist, or maybe even a doctor. So naturally he would choose biochem as his major.

But Amadeo doesn't like math. Ever since he was in the kindergarten or first grade, his teacher would ask him a math question, and more often than not she'd tell him that he was wrong. By the time he reaches the third grade, he's decided that he hates math. He is definitely what I'd call a "dren." When the teacher is giving a math lesson, Amadeo tunes out and counts down the minutes until it's time for another subject, such as his favorite -- science.

So we can see the problem here -- Amadeo is strong at science but weak at math. Bill tells us that if Amadeo wants to be a biochem major, he should have passes Algebra I/II in high school and be prepared to take Calculus I/II in college. As we already know, other traditionalists like SteveH would take it a step further -- since Amadeo desires a STEM degree like biochem, he should be taking Algebra I in eighth grade and AP Calculus as a high school senior. But surely these classes are for above-average math students, not below-average kids like Amadeo.

Amadeo manages to earn at least a C in his math classes since he's admitted to a four-year university, but Bill says that his teachers have lied to him (yes, just like the chairman in Doonesbury). Most likely, Amedeo passes his tests by learning just math enough the night before, then forgetting it the next day. But this is no excuse to Bill -- he states that he still remembers the Calculus he learned decades ago, so Amadeo ought to remember the Algebra in the few years between his high school classes and the university math placement test.

Instead, Amadeo fails the placement test, is placed into remedial math, and then fails the remedial class too. (Yes, this professor actually does give him a failing grade.) And so he chooses to drop Algebra in favor of a Statistics course, and changes his major to become a teacher.

Bill doesn't address this part of the story in his comments, but we can guess his reaction based on his previous posts. First of all, Bill doesn't like the idea of weak math students taking Stats. To him, a true Stats class is harder than Algebra II, and so the Stats class should be open only to those who earn A's and B's in College Algebra, not D's and F's.

Second, we see that Amadeo, who isn't good at math, wants to become a teacher. If he's trying to be an elementary school teacher, he must teach all subjects, including math. The fear is that Amadeo will pass his disdain for math to the next generation of Amadeos (and Raquels), who then likewise find their own paths to becoming scientists and doctors being blocked by weak math skills.

Perhaps Amadeo might want to become a secondary science teacher instead. To teach high school science classes he'd likely need to be stronger at math, but perhaps he could get away with teaching middle school science. As a middle school teacher, he'd be able to focus on the science -- and perhaps he could give a nod to the connections between science and math. Then maybe his weaker math students might be drawn to the path to a STEM major. They could see why it's important to learn math if they are interested in science.

It would be great if every middle school science teacher could demonstrate the connection between math and science in order to enlighten students. But I can tell you of one middle school science teacher who failed to make this connection last year -- and that teacher is yours truly.

Last year, I remember one seventh grade girl who told me exactly what she wanted to be when she grows up -- a veterinarian. I still remember one day when I thought I heard her having a lewd conversation about private parts with other students. It turned out that she was actually talking about the private parts of a dog. In other words, she was talking about her dreams, her future.

But this student, just like our fictional Amadeo, is a weak math student. I remember often when I gave her a "Monday Five" worksheet (see my February 13th post for more info). I remember how much she'd struggle with multi-digit addition problems, and she complained about being required to do them. Again, she believed that only smart "nerds" do multi-digit addition without a calculator, and ignored me when I told her the opposite -- that only "drens" need a calculator for multi-digit addition.

I can easily seeing this girl meeting the same fate as the fictional Amadeo and the real Raquel. She tries to major in biology or a STEM major, fails math, and is no longer able to take the classes needed to become a vet.

I could have told her that she needed to learn more math in order to become a vet. But this would have failed, because she could have retorted that surely she needed to learn more science in order to become a vet -- especially the life science taught in seventh grade under the California Standards. But I failed to live up to my end of the bargain and teach enough science last year, except a little to eighth grade (and hence almost none to the seventh grade).

Yes, this is going to turn into yet another post where I write about last year's science failure. As today is still Mole Day, I wonder whether I should have given a Mole Day lesson last year. I admit that I was considering it last year, but in the end I didn't. And so for the rest of this post, I wish to imagine what my Mole Day lesson would have looked like.

One reason I didn't give a Mole Day lesson last year was the fact that October 23rd, 2016 happened to fall on a Sunday, a non-school day. Then again, it's not as if it would have been any better this year with Mole Day on a Monday -- as in Coding Monday (provided, of course, that computer day is still scheduled for Monday this year).

Of course, in years when Pi Day falls on a Sunday, we observe it in school the previous Friday, so it's possible that I could have done the same with Mole Day last year. (It's just that I was already noncommittal about Mole Day last year, and the Sunday thing put skipping it over the top.)

But let's imagine that I really did observe Mole Day last year. I've written earlier that my math and science units should span the full week, not just Friday. So let's write out an entire Illinois State week for the days leading up to Mole Day:

Monday, October 17th: Coding
Tuesday, October 18th: Math Traditional Lesson
Wednesday, October 19th: Learning Centers
Thursday, October 20th: Science Project
Friday, October 21st: Math Weekly Assessment

First of all, notice that last year, the Mole Week project would have been for eighth grade. That's because last year I would have taught to the old California Standards, with physical science as the eighth grade focus. This year, I would have switched to the NGSS, and chemistry appears in seventh as well as eighth grade. (So my future vet would have celebrated Mole Day this year instead.) Of course, "moles" don't actually appear until high school, but it's a good idea to introduce them if we're going to have a Mole Week.

Now what should the math lesson for this week be? The week of October 17th is the eighth week after the Benchmark Tests, and so with one standard per week, this is the eighth standard. There are two standards in NS (Number System), so we should be in EE6 (Expressions and Equations):

CCSS.MATH.CONTENT.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

(By the way, in my last post I mentioned Kate Nowak and the textbook she was writing. On her blog, she mentions how she teaches similar triangles before slope in order to meet this standard. But we must follow the naive order where all EE standards appear before any G standards.) Anyway, this standard has nothing to do with moles.

Normally, the math standards and science standards don't line up. One big reason is that the NGSS doesn't divide the middle school standards into grades -- the states do that. So it's impossible to write a text in which the math and science standards correspond to each other. This is a shame, since we want to convince the Amadeos and Raquels and future vets of the world that it's helpful to learn math if they want to major in science. So it would help if the science lessons allowed the students to apply the math they're currently learning in the math classes.

Still, it would be great during a special week like Mole Week to make the math and science lessons actually correspond. So what math standard would fit during Mole Week? Hey, that's easy -- Mole Day is 10/23 because Avogadro's Number is 6.02 * 10^23. This number, 6.02 * 10^23, is of course written in scientific notation. And hey -- there's a scientific notation standard in eighth grade:

CCSS.MATH.CONTENT.8.EE.A.4
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology

This standard, EE4, is only two off of EE6, the prescribed standard for this week. So it would be easy to slow down the lessons so that EE4 is covered during Mole Week instead of EE6. (We'd just have to make it up later on so that G8, the last "major content" standard, is reached before the SBAC.)

I've stated before that the Friday assessment should based on the previous week's standard in order to give the students time to learn it. So the assessed standard would be EE3:

CCSS.MATH.CONTENT.8.EE.A.3
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.

Because of this standard, it's better to write Avogadro's Number as 6 * 10^23 (that is, a single digit times an integer power of 10) and save the mantissa 6.02 for high school. Students can then solve problems using Avogadro's Number, such as "How many atoms are in two (or whatever) moles," and similar questions. Tuesday's math lesson can begin with such problems on a worksheet.

There were a few things happening at my school during the week of October 17th. Wednesday of this week was an Illinois State observation, but the observed class was seventh, not eighth, grade.

More importantly, there was the California Earthquake Drill on Thursday. This drill always occurs on the third Thursday in October, and in theory the time should match the date -- that is, last year it was on 10/20 at 10:20 (and this year it was 10/19 at 10:19). In reality, the drill occurred at our school at 9:00 in order to avoid elementary recesses. (California state leaders should have anticipated that elementary schools would want to avoid recess. The drill should have been the third Thursday in September instead of October, so that the corresponding time would be in the 9:00 hour, before most schools have recess.)

But I arrived at school that day believing that the drill would be at 10:20 -- in the middle of eighth grade math class -- instead of 9:00, in the middle of seventh grade math. So that week, I would have made a lesson plan based on a 10:20 drill and having only half a class that day. Instead of a science project, I might have given the EE3 assessment instead. It's also possible that, with the previous Wednesday (October 12th) being Yom Kippur, this would mean two fewer school days between the introduction of EE3 and its assessment. I might have used this as a reason to give a Dren Quiz instead of the EE3 assessment.

Then Friday could be the science activity instead. Indeed, I like the idea of having a Mole Day party on October 21st as well, and it may be better to have a party on the same day as a project rather than an assessment. It actually depends on what exactly the science project is. In theory we should be using Illinois State projects only -- and there are some chemistry projects near the front of the Illinois State physical science text.

Provided that I gave a Illinois State project the previous week (meaning that I'd have fulfilled the every two-week requirement), I could create my own Mole Day project. For example, I could have the students pour 1.2 * 10^25 molecules of water into a beaker. I'd give them the information that one mole of water is about 18 grams or 18 milliliters, and so they need to pour 20 moles or 360 ml. (A crude approximation, by the way, is that 10^24 molecules of water is one fluid ounce.) I could also ask them the reverse -- pour a certain amount of water into the beaker and then figure out how many molecules it is. Notice that the standard allows students to use calculators ("scientific notation that has been generated by technology"), but they should avoid calculators during the water project in order not to get the calculators wet.

There was enough time to give this project before the 10:20 drill -- and even more time once I found out that the drill was really at 9:00. So our week would look like this:

Monday, October 17th: Coding
Tuesday, October 18th: Math Traditional Lesson (EE4, scientific notation)
Wednesday, October 19th: Learning Centers (incorporates math and science)
Thursday, October 20th: Science Project (counting molecules of water)
Friday, October 21st: Math Weekly Assessment (on EE3), Mole Day Party

Again, this was the week leading up to Mole Day Sunday last year. Some sources define Mole Week (or Chemical Week) as the Sunday-to-Saturday week containing Mole Day, and so Mole Week was  actually the week of October 24th last year. Giving the Mole Week project one week later means delaying EE4 another week (so one less week to reach G8 by the SBAC). There was no earthquake drill or observation that week -- instead there was a meeting with an Illinois State consultant. There might have been more pressure on me to give an Illinois State project that week. In reality, I gave the lesson H2O + ? from the math STEM text that week -- a project without any actual H2O involved. So I feel justified in giving that project and then adding my own water project to it. Of course, I'd send only photos of the original project to Illinois State.

Sarah Carter, one of my favorite teachers to link to, is now a Chemistry teacher in addition to being an Algebra I teacher. So Mole Day would be the perfect holiday for her to celebrate. Unfortunately, there's no mention of Mole Day on her blog today (or of Hexaflexagon Day for that matter). Instead, she posts her weekly Monday Must Reads.

One of these links, though is a teacher (Katherin O'Hara) who actually gives the Monty Hall problem in her class. After talking about Monty Hall all month, I feel I should point this out. Unfortunately, O'Hara provides only a Twitter link, and I don't post link to Twitter here -- especially when the Tweet is already seven weeks old.

OK, let's finally get to the U of Chicago text.

Lesson 4-7 of the U of Chicago text is called "Reflection-Symmetric Figures." (This corresponds to Lesson 6-1 in the modern Third Edition.) Two years ago I skipped Lesson 4-6, which affected the way I covered 4-7 as well. So this is what I wrote three years ago about today's lesson:

Section 4-7 of the U of Chicago text deals with reflection-symmetric figures. A definition is in order:

A plane figure F is a reflection-symmetric figure if and only if there is a line m such that r(F)=F. The line m is a symmetry line for the figure.

In other words, it's what one usually means when one uses the word "symmetry." Some geometry texts use the term "line-symmetric" instead of "reflection-symmetric." Some geometry and algebra texts use the term "axis of symmetry" instead of "symmetry line" -- especially Algebra I texts referring to the axis of symmetry of a parabola. Some biology texts use the term "bilateral symmetry" instead of "reflection (or line) symmetry" - in particular, when referring to symmetry in animals. As animals are three-dimensional, instead of a symmetry line there's a sagittal plane.

Indeed, it is this last topic that makes symmetry most relevant and interesting. Most animals -- including humans -- have bilateral symmetry. I once read of a teacher who came up with an activity where the students look for the most symmetrical human face. The teacher blogged about how students who are normally indifferent to geometry suddenly came fascinated and engaged to learn about the relationship between symmetry and human beauty. Unfortunately, this was more than a year ago, and I can't remember or find what teacher did this activity -- otherwise I'd be posting a link to that teacher's blog right here!

In the Common Core Standards, symmetry is first introduced as a fourth grade topic:

CCSS.MATH.CONTENT.4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Later on, symmetry appears in the high school geometry standards:

CCSS.MATH.CONTENT.HSG.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Notice that if a reflection over a line carries a polygon to itself, then that line is a symmetry line. But symmetry lines for polygons formally appears in Chapter 5 of the U of Chicago text. Right here in Chapter 4, we only cover symmetry lines for simpler shapes -- segments and angles. The text reads:

"In the next chapter, certain polygons are examined for symmetry. All of their symmetries can be traced back to symmetries of angles or segments."

For segments, the text presents the Segment Symmetry Theorem:

A segment has exactly two symmetry lines:
1. its perpendicular bisector, and
2. the line containing the segment.

The text gives an informal proof of this -- as the mirror image of an endpoint, there can only be two possible reflections mapping a segment AB to itself. One of them maps A to B and B to -- and that mirror must be the perpendicular bisector of AB, by the definition of reflection. The other reflection maps A to A and B to B -- which means that both A and B must lie on the mirror, since the image of each is itself. No other symmetry is possible. QED

But we also want to work with angles. The first theorem given is the Side-Switching Theorem:

If one side of an angle is reflected over the line containing the angle bisector, its image is the other side of the angle.

An informal proof: the angle bisector divides an angle into two angles of equal measure. The picture in the U of Chicago text divides angle ABC into smaller angles 1 and 2. Now the reflection must map ray AB onto a ray that's on the other side of the angle bisector BD, but forms the same angle with BD that AB does with BD. And there's already such a ray in the correct place -- ray BC. Notice that part b of the Angle Measure Postulate from Chapter 3 already hints at this -- the "Two sides of line assumption" gives two angles of the same measure, one on each side of a given ray. QED

The other theorem, the Angle Symmetry Theorem, follows from the Side-Switching Theorem:

Angle Symmetry Theorem:
The line containing the bisector of an angle is a symmetry line of the angle.

Earlier this week, I wrote that we'd be able to prove the Converse of the Perpendicular Bisector Theorem after this section. As it turns out, the Side-Switching Theorem is the theorem we need.

Converse of the Perpendicular Bisector Theorem:
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Given: PA = PB
Prove: P is on the perpendicular bisector m of segment AB.

Now I was considering giving a two-column proof of this, but it ended up being a bit harder than I would like for the students. But as it turns out, even though the U of Chicago text doesn't prove this converse, in Section 5-1 it gives a paragraph proof of what it calls the "Isosceles Triangle Symmetry Theorem," and the proof of this one and the Converse of the Perpendicular Bisector Theorem are extremely similar. After all, we're given that PA = PB -- so PAB is in fact an isosceles triangle!

Proof:
Let m be the line containing the angle bisector of angle APB. First, since m is an angle bisector, because of the Side-Switching Theorem, when ray PA is reflected over m, its image is PB. Thus A', the reflection image of A, is on ray PB. Second, P is on the reflecting line m, so P' = P. Hence, since reflections preserve distance, PA' = PA. Third, it is given that PA = PB. Now put all of these conclusions together. By the Transitive Property of Equality, PA' = PB. So A' and B are points on ray PB at the same distance from P, and so A' = B. That is, the reflection image of A over m is B.

But, by definition of reflection, that makes m the perpendicular bisector of AB -- and we already know that P is on it. Therefore P is on the perpendicular bisector m of segment AB. QED

Let's think about what we're trying to prove here. We want the Converse of the Perpendicular Bisector Theorem -- and consider what I wrote earlier about the proof of converses. The proof of the converse of a statement often involves the forward direction of the theorem and a uniqueness statement -- and even though we didn't use the forward direction of the theorem here, we did use a uniqueness statement here. As it turns out, given two distinct points A and B, there exists only one line m such that the mirror image of A over m is B -- and that line is the perpendicular bisector of the segment AB. And so if we can somehow find out another way that the mirror image of A over m is B, we'll have proved that m is the perpendicular bisector of AB. So that's exactly what we did above -- we proved that a certain line (the angle bisector of APB) is the perpendicular bisector of AB.

In this section, we found symmetry lines for simple figures such as segments and angles. But can we find symmetry lines for the simplest figures? As it turns out, a point has infinitely many lines of symmetry -- any line passing through the point is a symmetry line. But a ray has only one line of symmetry -- the line containing the ray.

Finally, does a line have a line of symmetry? This is exactly the answer to Question 25 of this section, in the Exploration/Bonus Section. A line -- considered as a straight angle -- contains more than one symmetry line. This is because any point on the line can be taken as the vertex of that straight angle. Since straight angles measure 180, their angle bisectors must divide them into pairs of 90-degree angles. Therefore, any line perpendicular to a line (straight angle) is a symmetry line of the given line. This is what I called the Line Perpendicular to Mirror Theorem. It implies that a line (straight angle) has infinitely many symmetry lines. (Of course, the line has one more symmetry line that I didn't mention -- namely the line itself.)

I included Question 24, even though it appears to mention corresponding and same-side interior angles formed by two lines and a transversal. But nowhere in the question does it mention anything about the two lines being parallel.

I left out Questions 16 and 17, which give the construction of an angle bisector. I finally plan on going to constructions sometime next week. But here's another video from Square One TV, where doctors have to perform a "bisectomy" on an angle. (Unfortunately, only the entire 30-minute show is available on YouTube -- the "bisectomy" doesn't begin until the 11-and-a-half-minute mark.)