 So, we want to look at a couple of weird congruency examples. Shania, the first one, if you draw yourself a right angle triangle like this, let's say, and then try and draw an identical one, same size, same shape right next to it. Courtney, because I can't draw to save my life, I'm going to cheat. I'm going to go like this. I'm going to go copy. I'm going to click here, and I'm going to go paste. Try and make them look as identical as possible, because Joe, they're going to end up being congruent. Even though they're not yet, because I don't have enough information yet. What kind of a triangle is this, Marcus? This is a special triangle. It is a right angle triangle, okay? And here's what I'm going to tell you. I'm going to tell you that this right here is five long, and so is that. I'm going to tell you that this right here is three long, so is that. And I'll call this triangle A, B, C, D, E. So copy that out. Now remember, Alex, we can show side lengths either with numbers or with symbols, because the other thing I could have done is I could have done about one, two, three, one, two, three, one, two, one, two, either of those is a way to show them the same way. Problem is this. Right now, I have side angle, or if I go in the other direction, right now, all I have is ASS. This is not enough for congruency right now. If I knew that angle, it would be side angle, side, yay. Or if I knew angle, side angle, if I knew that angle. But Marcus, what kind of a triangle is this, remember what I said? Only because this is a right angle, there's one more thing that I know about this. It's a theorem. It's a rule. It begins with letter P, and it rhymes with Egerus. You see, I also, as soon as it's a right triangle, I also know this, and we're going to write this down. We know A squared plus B squared equals C squared. You have a blank half page on the notes from last day, right? Yeah. This room. We're going to be doing two diagrams, so don't do it a whole page, so I'll figure it out. Sierra, this is C. The C was always the hypotenuse. How big is C in this triangle? How big in this triangle? Same size? And this is little A. How big? How big? Yeah, so here's what we're going to see if you can reason this C if you guys could go up with this. If I know that C is the same in both triangles, and I know that A is the same in both triangles. Is there any conclusion that I can arrive about that number right there? Adam, you're nodding or are you just twitching because you're hearing music in your head? Is that a nod? What can I conclude? And not only that, look, if that's the same in both triangles, and that's the same in both triangles, you know what you know about this in both triangles? It's also, I know because of Pythagoras that those two are the same length. If A squared plus B squared equals C squared, Simon works on the first triangle, it's got to work on the second triangle. The only way it can work on the second triangle is if B is the same number or the same length in both. Is that okay? And now, hey, which congruency rule do I have? Do you see it? I got two, side, side, side, or side, angle, side. Let's make a little note here, Pythagoras. And now, let's write a congruency statement, triangle A, B, C, and the symbol for congruency was the equal sign with a little curly capital N. And then I said the second one, the order matters. Taylor, what matches A in the second triangle? Yep, what matches B in the second triangle? D is a dog. So this is an example, Jordan, where you have to add a piece of information. But because it's a right angle and you know that Pythagoras works, you can say, oh, I do know that third side has to be the same. In fact, on your yellow reference sheet, I'm pretty sure on that back page, so here was our three congruency rules, and then right next to it, on the right hand side, property number two, if two sides of a right triangle are equal to two corresponding sides of another right triangle, then the third side has to be equal to the property of Pythagoras and you've got side, side, side in disguise. Did I include the reason? That's the first weird extra one, Aaron. The second one, once again, we're going to draw a different triangle, not a right angle triangle. Try and draw something sort of like this. Oh boy, that's terrible. Again, one act, one take two. It's not isosceles. It may look like it. It's not meant to be. And then try and copy that same triangle out again. Mr. Dewick will cheat because he has no artistic skill whatsoever. And let's call this triangle Q-R-S and how about X-Y-Z. So far so good, Emily. Here's the information I'm going to give you. Put a little X right there, little X right there. Those two angles are the same. Put a little checkmark right there, a little checkmark right there, those two angles are the same. Put a double hash mark on this side, a double hash mark on this side, those two sides are the same. Is this angle side angle? Well no, because the side had to be between the two angles. This is not angle side angle. Back, you know what this is? This is angle side, which isn't a rule. The three rules, Sam, that I gave you were side, side, side, side, angle, side, angle, side, angle. Is this congruent? We have to make one more logical step. Ready, Sam? Are these two angles the same? Are these two angles the same? What does every triangle add to? If these two angles match, and these two angles match, what can you tell me about these two angles? Why do they have to match? Because if this plus this plus mystery adds to 180, and this plus this plus mystery has to add to 180, I think the two mystery angles got to be the same size. We can logically assume, we can say, you know what? This angle, and this angle. We can add that piece of information to our diagram, and say, now which congruency rule do I have? Do you guys see it? Angle angle side isn't a congruency rule. Matt, angle side angle. So now I can say, triangle Q, R, S is congruent to triangle Q goes with Z, R goes with Y, and S goes with, Mr. Dewick, it would be nice if you put that one on our yellow sheet too. Oh, I wonder if I did. Let's see. Oh, note number one, if two angles of one triangle are equal to two angles of another, then we know the third angles will be equal, which means you can add that to your diagram, and you might now have a congruency rule. Might not. Might still need more information, but Emily, you might. And the rule we said was angle here, side here, angle here. So those are the two ones I wanted to talk about. Can you take a look at the homework, please, this big package which I assigned starting out, I think we assigned all the way up to S22. What I would like you to do, I'm going to do two or three examples with you and the rest of the class is yours to work. We're basically now pretty much done the unit. On Tuesday, you can expect another take home quiz. I'll try and make up some kind of a unit review assignment as well. Right now, I'm thinking your test is not going to be next week, it's probably going to be the week after. That's going to be on your test. Very similar to quiz number two, the one that you marked yesterday, and two proofs. One proof on congruency and one proof from stuff that we've done before. So what I'd like you to do now that you're on S22, turn the page. I'd actually like you to go to page S23 and find number six, this bad boy right here. Danielle, this is actually a completely unguided proof. You'll notice over here they still have blanks for you to fill in. Here, they haven't given you any blanks to fill in. I'm going to give you a couple of those to try. On your test, I'll give you blanks to fill in. But Jordan, I figure if you can handle the unguided proofs, you'll find the guided ones where it's filling in the blanks way easy. That's the theory. So you ready? We're going to do number six together. Sydney, what am I trying to show? What does it say here? Can you read that to me, please? Triangle, P-A-T is congruent to triangle, T-O-M. How am I going to do that? Side angle, side, side, side, side, or angle, side, angle? I don't know which one. How am I going to do that? I'll let the givens tell me. And the very first thing, Joe, that I'm going to do is I'm going to write down the first given. Read me the first given. We're on number six now, Joe. Help me out, Jordan. How do I know? Given. And let's mark that on our diagram now, too. What's the symbol for a 90-degree angle? That little half-square thing, right? Angle A, angle O are 90 degrees. And right now, I think to myself, in terms of my congruency rules, Aaron, I'm probably going to use one that has angle in it. Is there anything else I now know? I don't think so. So let's move on. Danielle, can you read the next given to me, please? P-T, double hash mark equals T-M, double hash mark. Oh, now I have side, which may come in handy. Adam, what's the next given? A-T, thunk equals M-O, thunk. Oh, I have side, side angle. Or, oh, you know what? Right now, all I have is angle, side, side. Is that a congruency rule? No. Marcus, what kind of triangles are each of these two? They're right angle triangles. What do I know about right angle triangles if I have two sides matching? What can I say? I have no idea what you said. I heard, oh, you can, I heard anything with an S. So what can I say here? Ah, that's why they gave me these three things. My next line, I'm going to write AP is the same size as what? T-O, why, Marcus? I think I would, except I'll probably abbreviate it as Pyth. I'll mark that on my diagram. 1, 2, 3, 1, 2, 3. And ha-ha, which congruency rule do I have? In fact, I have two of them. Now, I have side, side, side, or side angle side. So I can now go to my final statement, which is always the show. I'll even do it in red so it stands out. Triangle P-A-T is congruent to triangle T-O-M, side, side, side, or if you did side, angle, side, I'd give you full marks, too. That's approved. Shut up, judge. Guilty. Sit down. Or innocent, if you want to be a defensive term, but no room for argument there. Ironclad proof. Two more of them. Let's take a look at this one here, number 10. Now, let's do number 11. I saw number 10 had a right angle. It's going to be quite digress again. Let's try number 11. Boston, what am I trying to show? There's a symbol in front of it. No, that's not an angle symbol. I'm pretty sure it's a triangle in front of it. I want to show that triangle hog and triangle dog are congruent. How? Side, side, side, side, angle, side, angle, side, angle. How? I don't know. Courtney, let's let the givens decide. Courtney, read me the first given, please. Let's write that down. And let's mark that on our diagram. Those two are the same length. I just saw something. In fact, you know what? I just figured out why they gave me that piece of information. What else do I now know? Do you see it? What? What's isosceles? The great big triangle is isosceles. Yes? What else do you know, then, if it's isosceles, that great big triangle? Not only are the two sides the same, but what else do you know, Shania? I'll bet you that's what they want me to write next. You see how I figured out by saying, why did they tell me this piece of information? And for an unguided proof, that's what you want to be doing, Shay. Any time you write down a given, you want to ask, what else do I now know? I better write that down. And it's going to be angle H equals angle D. Isosceles triangle. This angle and this angle are the same. In Shania, I now have, look up, either side angle side, if they tell me those two are the same, or angle side angle, if they tell me those two are the same. I'm very close to a congruency rule. Let's see. Sierra, have I used up all the givens yet? Well, can you read to me the next given, then, please? Angle, how about a D, Mr. Duke? D-O-G, that's given, and now let's label that. H-O-G, that's that guy. D-O-G, that's that guy. Now, right now, Jordan, if I start here and go this way, I have either, or if I start here and go this way, I have that. Are those congruency rules? No. Wait a minute, though, wait a minute, wait a minute. Hey, hey, hey, hey, hey, hey. Alex, are those two angles the same? Are these two angles the same? What can you tell me about these two angles, then? They're the same? Why? You know what? This is that rule number one on the back page, on the right-hand side of the back page, at the third angle is in the triangle one. Yes, so you know what? Let's go like this. I'm going to call this angle one and angle two, because I'm lazy and don't want to write all letters. I'm going to say angle one equals angle two. Recall that third angle in the triangle. If two angles match, the third one has to. And I'll even symbolize that by putting a little arc in the hash mark, a little arc in the hash mark. I would take that. If you said angle sum of triangles, because you know they add 280, I would accept it. I'm being proper here and saying, but really, the argument is the fundamental party argument is saying, oh, and I also know, two angles match. So the third one does. But yeah, by the way, Marcus, can you see which congruency rule just popped out? Yay, we're on our last line. Final statement, triangle hog is congruent to triangle dog because we force fed angle side angle. Am I going to give you an unguided proof on your test? No. Well, I give you a guided one. You see the difference between unguided is all blank, no hints. Guided is, hey, they gave me some lines to fill in. I'll give you one something like this and one will be a congruency proof and one will be like from lesson five, okay? You guys want to do one more or do you want to just start whacking away at some of these? You know what? I'll let you start whittling away at some of these and I'll start out next class doing a couple more. What's your homework? Right now, I would say, first of all, if you have not yet and some of you haven't, hand it in geometry package number one or geometry package number two. Those are absolutely due today. I'm marking everything this weekend and sending out an update. Also today, I would love it if you have lesson five, Proving Conjecture's homework, which was our first introduction to some of these proofs. If you have finished all of those on the congruency homework, so so far I had told you to try up to S22. I think you can now try see how far you can get up to about S16 geometry 10. That's not due for Tuesday, but on Tuesday, I'm leaving you more stuff if you're trying to pace yourself. That'll keep you so you're not having to do 10 hours of homework in one night. Does that make sense? I don't want you to have homework for the long weekend, so here's the deal. Work hard this class and I don't care how far you get. If you're not this far on Tuesday, but I remember that you worked hard, Jordan, I'm good with that. It's got a long weekend. I don't like giving up homework on a long weekend. Goof around and I'll probably have you in after school last block on the Tuesday getting caught up. Don't make me do that. Let me hit save.