 Lesson five, cruising conjectures using angles. First, a little history. This first page, this is just the nerd within me telling you where some of this came from. I'm not going to be asking you this on a test. Let me turn your brain off. Read along with me. Euclid, who was a Greek mathematician around 300 BC, around 2,300 years ago, published a book, and the title of the book was called The Elements. Not a very, very exciting sounding title, but it was such a good textbook. It's actually still being used, translated into English or French or Italian. It was phenomenal. It was ahead of its time. In fact, at one point it was the most published book in history after the Bible. Why was it so revolutionary? What Euclid did for the first time, and nobody had really done this in a formal way before, he said, if I know these few things, what can I prove from that? Using only logic and reasoning, not using religious faith, not using any steps where I'm just making stuff up. What can I prove using this and this alone? No one had really thought of doing that yet. In it, first of all, he started out by proposing what he called Tanner five common notions. What he really meant by this, Nicole, is that this seems obvious to me. I'm not going to prove this. These are the five. Things that are equal to the same thing are also equal to one another. Say what? If A equals B and C equals B, then A equals C. So I'm not going to prove that. That seems reasonable to me. If equals are added to equals, then the holes are equal. Say what? If you add cars to cars, you get cars. If you add Xs to Xs, you get Xs. He said, I'm not going to prove that. That seems reasonable. If equals are subtracted from equals, then the remainders are equal. Same idea. Things that coincide with one another, equal one another. And the last one was, the hole is greater than the part. If you add things together, you get a bigger answer. I'm not going to try and prove that. That seems reasonable to me. With me so far? Then he gave what he called his five postulates, and these are fairly famous among math nerds. He said, let the following be postulated. These are the five taken directly from the Greek into the old English. It's kind of weird to read. I'm not even going to get into them, but there they are. And from those five common notions and from those five postulates turned into 13 volumes and thousands of pages. He asked, what can I prove starting with those? Big deal. Huge deal. It is. This approach inspired centuries of the same procedure, but to areas other than geometry, philosophy, politics. Our court system is a classic example where you enter items into evidence, postulates. You try and refer to previous cases, previous common notions. Our whole court system is based on that. Much of our political system is based on that. We call it debating. You may watch, if you watch the political channels, they're always debating in the house. They're trying to go statement reason to defend or refute an argument. For us, our binding case precedence will be the rules that we proved so far. The ones that are on the front part of your page, the yellow sheet. And some common notions, stuff that just seems obvious, guided proofs. Here's an example. Given this particular piece or court of law or case or evidence, and given this information, people's exhibit one, people's exhibit two, people's exhibit three, prove guilty. You hear the court system in there? Pick up your pencils. Here we go. I always start by asking, just like they would in the court of law, by trying to show or prove. Joel, what is it asking me to try to show? It wants me to show that these two lines are parallel. Did you say parallel? Yes? On your yellow sheet, right away, I'd be looking at my parallel line rules. There's three of them. I think what I'm going to try to do to convict this prisoner or show that this prisoner is innocent, depending on which side of the legal system you want to go to, is to prove that either corresponding or alternate interior or co-interior exists. How am I going to prove that, Tatiana? With the evidence I have to deal with. The first thing it says, can you read this to me, Tanner? This first statement here. Look up. And we're on this page. You with me now? Is that right there? The reason for that is it's given. Let's write that down. From now on, what would be a good abbreviation for given? What would be a good abbreviation for the word given? Anyone? What? C? Yes, I know. If you write a G, period, I'll clue it and give it. And I'm going to label that on my diagram right away as well. What did it say? It said that AC was the same size as BC. The hash marks on there, that was very nice. If I know that, what else do I know on the diagram? Well, right away I'd be saying, this has just become a special triangle. What kind of a triangle is this now? Maybe I'm going to use that in my court case. Let's see. The next thing it says, Zach, can you read this to me, please? This line here? Stop, there's something in front of it. Okay, angle. Stop, angle, B, A, C, that's that guy right there. Let's put a little tiny X there. Keep reading. A, B, C, let's put a little X there to show, we'll put the same symbol there to show those are equal. Oh, and it even says, and they equal how many degrees? Can I figure out how many degrees they equal? If that's 30, and I know from people's exhibit one that this is an isosceles triangle, how big does each angle have to be? 75? Objection, your honor, calls for speculation. Y, what reason can we put here? This is not a given one. We've introduced a previous case. How do we know these two are the same? You just said they're equal because they're equal. Your honor, it's an isosceles triangle, objection overload. Okay? So you know what? I could have put an X there, but I'm actually going to, and you can just write above the X, say, oh, that's 70 degrees, because I could actually figure out how big they were. Ashley, what's the next thing that wants me to do? Can you read that to me? Stop. That's that one. Is how big? How big is that angle right there? That's a 70. Oh, it's 75. I wrote the wrong angle. Thank you, Nicole, for me making a mistake. 75 degrees and 75 degrees, 75 degrees. How big is it, Nicole? Thank you. 105. How come? Well, no, we don't know that these two are the same yet. What did we just write down on the previous line? What are we introducing to evidence? The fact that these two here are each how big? 75 degrees. Why does that tell you how big this one is? What's this rule? Kayla, you're right. Nice and loud. Supplementary. So this is 105. Alexis, what's this next line? Can you read it to me? Nice and loud. That's this one. Yep. That's this one. Are these two the same? Why? Now, this one's so obvious. Same size. Here's what we've done. We knew this one was 105. With the evidence that they gave us, we were able to... Oh, actually, you know what? It's also given, isn't it? They told us DEF was 105. Give them. Not yet. Hang on. You're partly right, but not quite. The reason we can't say they're corresponding, have I marked any parallel lines on here anywhere? You see the little arrows on here? So I can't use those. In fact, what I'm going to do to prove that those two are parallel, I'm going to set up corresponding. In fact, that's what I've done. Jacob, what can you tell me now? What pattern is this in our parallel line rules? What you just said. So the reason that I know BC is parallel to DE is that I've force-fed corresponding into this court case. And once we've written that down, let's review the court case. Here's what we said. They gave us this diagram and evidence. People's exhibit one was AC equals BC. Given. They told us that. I labeled that on my diagram right away, the given. Then I said, why did they tell me that? Oh, I know those two angles reach the same and 75 because it's an isosceles triangle. Why did they tell me that? Oh, I know this angle is 105, supplementary. What does that mean? Well, these two angles are the same. I better point that out to the jury because I'm going to assume the jury's stupid. What does that mean? Oh, it must be parallel because I've just proved to you that corresponding exists. Let's try another one. Question two. What are we trying to show here? Can you read what it says next to show? Nice and loud. I'm trying to show that this length here is the same as that length there. Okay. How? I don't know. I'm going to let the givens kind of hint to me. So the first thing it says, Niki is this. Can you read this to me? How do I know? Oh, given. They already marked those sides the same. If they hadn't, I would mark those sides the same because I always mark at my diagram pole. Then it says P M N equals P N angle, P M N equals angle P N M equals how big? This angle equals that angle equals how big? He needs some help, Paul. Give me a little elbow. Is he going to make it? Angle. Let's keep going. So these two angles are the same. That's what this is. How big is each of these angles? Zach? How big? Oh, I thought you whispered it. Sorry. No? Well, how big? We need to know. 70, convince me. Ooh. So each of these, this is 70. This is 70. This is 70 because it's an isosceles triangle. Okay. Then it says angle LMP, angle LMP, angle LMP. This one is how big? Kayla, convince me. So let's put that on our diagram. Then it says angle MLP equals angle MLP. Angle MLP. This one, convince me. Tanner is alive now after a couple of hits to the head. Good. 35 degrees, 35 degrees because we called that angle sum of triangle. Sorry, there's a photocopy, the photocopy, the photocopy. The black dots come from. I can't get ready. Sorry. Okay. Then it says angle MLP, let's see, MLP, this guy, equals angle MPL, MPL, this guy. Do they equal each other? Yeah? I know. Oh, they gave me that. They gave me the 35. And it's also going to be an isosceles triangle, which is going to be helpful. Need to make this a little bit smaller to fit it all on one page. Can I? There we go. Then it says angle LM, sorry, not angle, line LM equals line PM. How do I know that this line is the same length as that line? Isosceles triangle. Yeah? And then the last line seems to be the conclusion. Now it's telling me, hey, you have now proved that LM equals PN. You have now proved that this line and this line, that this line and this line are the same length. How do I know? I don't think it's given. The shown is what I'm trying to prove. So that proved that, by the way, the show is always going to be your last line. Yeah. How do I know those are the same length? Let's walk through this again. Ready, Emma? Ready, ready, ready? And this time, let's keep an eye out for line lengths. I'm going to ignore some of the angles. Do I know those two are the same? So that means that this one and this one are the same. And the way down here, do I know that the two green ones, sorry, do I know that the change colors, Mr. Duick? This one, this line right here, and this line are the same. How does that help? Okay. What Paige said very quietly in lovely math nerdy terms is this. Look, look, look. Did we show that this line is the same? Let me erase some of these highlighted hash marks here. We already showed that this and this are the same. We did that right there. Yes? We already know that this and this are the same. We were told that right there. So if this equals that and this equals that, what can you say about these two lines? They have to be the same size because they equal the same thing. Each equals, let's see, LM equals PN, sorry, LM equals PM, PN equals, each one equals, here's the key statement right there. See it called? We showed that PN equals that and LM equals that. Well, if they both equal the same thing, they both equal each other. There is a fancy word for that. Give it to you because, you know, math nerd. If A equals B and C equals B, then if you know that and you know that, what can you conclude, Joel? The fancy word for that, transitivity. If you want to use it, it sounds impressive in a conversation. How do you know? Transitivity, idiot. Am I a nerd, Joel? Yes? Will I use that now? Yeah, I will. So when I say transitivity, it's this statement right here. Okay, what would a good abbreviation be for transitivity? Trans, okay. Couple more and I'm going to shut up for a bit. Example three. Example three and put it away right now, Rachel. Thank you. Welcome back. Colleen, see the great big shake they gave me? Circle and some other lines. What are they telling me to show? What are they asking me to show? Show what? They want me to show that these two lines are parallel. I think to do that, I'm going to force feed either corresponding or co-interior or alternate interior. How will I know which one to force feed? Well, the other information and the guided proof that they walked me through. Are you ready? Here we go. Sam, what's the first thing that they told me? Let's label that OB equals OA. How do I know those two have to be the same length? It's forming an isosceles triangle but I can't say that yet because right now I'm just trying to prove that it is an isosceles triangle. How do I know those two lines have to be the same length? Rob, the radii, which is the plural of radius and I'm going to use that from now on. One radius, two not radiuses, radii. Remember we said any radius has to be the same length as any other radius if you're in the same circle. Now, Nicole, it says angle ABO. ABO is that one right there. What angle is the same size as that one right there? Is that because she alive? Give her a little elbow for me. Oh, she's awake. Little daylight up chin. I don't know if you are. What's this angle the same size as? What other angle? This is what you were alluding to by the way. What did you say a few minutes ago and I said we can't assume that yet. Now that we know that those two sides are the same because they're radii, we can now say it's an isosceles triangle. ABO is the same as which angle? OAB or BAO. Those two angles are the same. I'll show they're the same by putting the same symbol in there. I used a dot or an X or a check mark or a star, the same symbol. Tasha, what's the reason that they gave me for the next thing? What's the reason that they gave me for the next line? Your hint is I'm running metal bouncy ball over the reason. Given. Oh, what's the given that they gave me? Okay, that goes here. Angle COB equals angle BAO. Oh, let's label that. So, angle BAO, this one. I'll put a little X there now. Let's make a little neater X, Mr. Dewick. Is the same as angle COB? Those two angles are the same. Now what? It says angle ABO, this one with the red dot, equals angle COB, this one with the blue X. How can I say that those two angles are the same? With me now, Emma. Ready, Emma? Watch, watch, watch, watch. Does this red dot equal a blue X? We already said that. Does this blue X equal a red dot? Then what can you say about this red dot and this blue X when I compare them to each other? Oh, candy for you, girl. Got to pause the video. I got to write that out. By the way, Sam, if you forget this stupid word because it isn't obscure and you write, they both equal the same thing. Full marks. Are we down to our last line? Yes. Then the show always goes on our last line. Apparently AB is parallel to OC. Oh, we just say these two angles were the same. Then it seems to me that I can put a red dot right there and I can put a blue X up there. Yes, because they're the same. Which parallel line rule have I forced that into here? Can you see it? Which one, Zach? What letter do you see? Z and Z, last letter of the alphabet. I'm looking for the one with the first one. Alt int. We prove that alt int exists. Therefore, they're parallel. Tatiana, are these tricky? Yeah. Am I going to put 10 of these on your test? No, probably two. But are these worth what we're really doing here? Tatiana is learning how to do an argument of several steps. Are you teenagers that like to argue? Yes. Do you suck at it? Yes. This is how you win an argument. Go cold. You pre-think it. You say, what are my reasons? What are my reasons? Then the other person has to be wrong. How many do I have here? Do one more. Sorry, what? I do. Too big. If we are at a staff meeting, for example, and we're debating a contentious issue, and we're trying to decide what the right thing to do is, I will make a couple of notes. And often I'll say, if I'm standing up, I may choose to say actually the point that so-and-so made. I think we can get around that or defeat that by simply adopting this. And so that takes that point out of contention. I really do when we're debating things, especially when most of you get older, you'll be at meetings where they're debating stuff. This is how it works. Volume is not an argument. Logic and reasoning is a good argument. Last one, Tanner, round number four. Paul, are you there? Yeah? Paul, what are they asking me to show? Show what? Thank you, Paul. They want me to show that these two angles are the same. How? I don't know. I'll let the givens and my brains tell me. What's the reason that they just gave me here, Joel? Oh, what's the given they gave me? And you'll notice because this is a simpler diagram, they're not going to three-letter angles. If it's a simpler diagram, you know an angle is that one. Angle E equals angle H. Do you say those two angles are equal? Let's put the same symbol. How about a little check mark and a little check mark? Or you can put a star and a star or a dot and the dot or an X or an X or if you're really artistic, you can draw a little unicorn or a little unicorn. I'm not going there. David, what's the next reason they gave me? Where are the only sets of vertically opposite angles in this diagram? What were we looking for for vertically opposite letter? Do you remember? Ah, you know what my vertically opposite angles are? These two. Sorry? Those aren't vertically opposite. Those are in the corners. My vertically opposite angles, Nicole, are these two. But do we know they're the same size? Let's go like this. I'll go red X, red X. What will I write here? Angle E F J is the same size as angle H F G. Those two angles there have to be the same size. David, how do I know those two angles, even though there's no numbers, have to be the same size? Vertically opposite. Ready? Are these two angles the same? Are these two angles the same? What does every triangle add to? So what can you tell me about the size of those two angles if that angle matches that angle and that angle matches that angle? What can you tell me about these two? Why do they have to be the same? Because they have to be the same because they have to be the same. Here's what we're going to say. First of all, they are the same. We're going to say the third angle in a triangle. If two angles match in two triangles, the third angle has to match because why triangles add to 180? If you wrote here triangles add to 180, I'd probably give it to you as well. These are tricky. On Wednesday, I'm going to do these next three are fairly tough and also introducing some new ones to you. For now, I'm going to say this. Try number one, which is a bit further on attached to the same assignment. Try number two, try number three, try number four, and I have a take home quiz for you.