 So we're not marking question 12, question 11, and for some reason question 9, I just cut and paste from a different document. I didn't bother going in order. So this quiz is going to end up being worth 4 plus 5 plus 3, 4 plus 5. It's going to be out of 12. But we're going to start out for your own edification by going through these first ones here. Are you ready? Mark your own with me. See how you did. Now we're trying to show that this triangle, I think that's this little guy here, is congruent to this little guy here. Since that's what I'm trying to show, Marcus, I can already tell you what the last line is going to be. It's going to be that right there, although they wrote AOE and COF, but the same idea. And there's going to be a congruency statement. In other words, see this proof here? I knew that was going to be my last line. I don't know whether it's going to be side, side, side, angle, side, angle, or side, angle, side. See this one here? I'm just going to already fill it as last line. Triangle DGF is congruent to triangle FE, because that's what I was trying to show. Whatever you're trying to show, Jordan and the proof, that's always your last line and why. The rest of it is a little more hit and miss. I agree. So let's see. Well, it says AB is parallel to DC. Why, Joe? Now, I'm going to stop right now and I'm going to say, did I just write something down? I should mark that on my diagram. I was yelling at you about this yesterday. How do I show parallel lines? Here's how I don't show parallel lines. Boston, this does not mean parallel. It means same size. How do I show parallel lines? Arrows pointing in the same direction. It says that this line is parallel to that line and that's how I show it. What do I now know because those two lines are parallel? Now, now that I know those two lines are parallel, Taylor, I see a Zed, which makes me think that's what they want me to write here. Not sure, but I think so. So I'm going to say this angle and this angle are the same. Why are they the same? What reason? So that's my reason, half mark for that. And then the angle. Now, there's several ways you can do it. If you put like a little one there and a two there and just said angle one equals angle two, you could even say angle A equals angle C because those are the only angles there. Most of you probably want angle, let's see, E A O is the same as angle F C O. One mark for this line, half mark, half mark. If there's two blanks to fill in on one line. This line's worth one mark if there's two spaces, a half mark for each space. And to show that these angles are the same, there's several ways you can use a little loop and then put like a single hash mark that means of the same size or you can put the same symbol. Sometimes I put a dot there and a dot there or a star there and a star there or a check there and a check there. But some way if you write it down, Boston, put it on your diagram. There is no way you can keep track of all this in your head. Boston, what's the next thing? Read it to me. Look up. What's the next thing? Louder now. From here to here equals from here to here. How do I show the same lengths, hash marks? Why? This is what you were starting to ask before I cut you up. How come I know those two are the same length? Wouldn't you write folks? Yeah. The radius is. Oh, you even used the plural. By the way, traditionally if we want to show you that the point is in the center, we'll call it point O. Why? Because what does the letter O look like? A circle. So traditionally we've used O for the center of a circle to kind of let you know that's supposed to be the center. Just for what it's worth in case you're wondering. And I'm looking for congruency rules. Boston, right now, here's what I have. Angle side, something, or side angle, something. I don't think I'm using side, side, side. In fact, I kind of cheated and I glanced down here because this I'm allowed to look for hints. What rule, Adam, do they want me to use? So that must mean I'm going to end up going angle side angle. They want me to show that these two angles here are the same. How do I know that these two angles here are the same? Marcus. Ah, brood off. And let's call it angle E-O-A, or you could go A-O-E equals angle F-O-C, or C-O-F. And I'll mark that, Sierra, on my diagram. I forced myself never to move a line on until I've written it on my diagram. How will I show those two guys are angle? How about putting a check mark and a check mark that works for me? Angle side angle has now been proven. So give yourself a score out of four. If you're not sure about something, put a star next to it and lawyer with me afterwards. Now, I just need to remind you, I've said the last three are going to be for homework and counts. So one, two, three, stop here just so I don't accidentally do those questions. Just curious how many you got? Four out of four? So some of you? Good. Those that didn't? The nice thing is, it's pretty tough to get zero on these. I think most of you will get some of them blanks and it's really getting perfect is tough. I know. Oh, Simon, am I putting like 10 of these on your test? What do you just see? In fact, one congruency like this and then one from the previous lesson, lesson five, where we first started doing these kind of proofs and you're trying to show two lines of parallel or something. Question two. Question two. Devon, what am I going to write here? Yeah, you do. What's the rule? So what am I going to write here? Oh, someone hasn't watched the videos since she's okay. We need to chat. You know, Matt, what am I going to write here? The thing they gave me. That's what given means. Yeah. JL equals NL. Courtney, did I just write something? I better mark it on my diagram. JL, this side right here is the same length as NL. Remember, for lengths, for lengths, Aaron, we said using your textbook. How does that help me? Does that tell me anything new? I don't know yet. Jordan, how many givens did they give me? I've used one. Do you see a given appearing anywhere else? Do you see a given appearing anywhere else right now? Then you know what? I think it would be reasonable to assume that I think on the next line is my next given. You see how I kind of use logic and reasoning to actually figure that out, too? So hey, Jordan, what is the next given? Read it out to me. Angle J equals angle N. Angle J, that's this guy here, equals angle M, that's this guy here. And Shania, now I think either side angle, side, or angle side angle. If I went side, oh, wait a minute, Shania, what's the rule that they wrote here? I'm looking for an X. Ah, you know what? They want me to say that those two angles are the same, don't they? That's the vertically opposite angles. They want me to say that angle J, L, K is congruent to triangle, Mr. Dewick. Congruent, Mr. Dewick is the same size as, sorry, angle N, L, M. And again, you could have said K, L, J and M, L, N as long as the L is in the middle for both of them. Yes? Which congruency rule have I force fed into here? Which congruency rule have I force fed into here? Because this is my last line. It's either going to be side, side, side, side, angle, side, angle, side, angle. Which one have I force fed into here? Turn right, see it? Angle, side, angle. So one mark for that, half mark, half mark, one mark, one mark. There's your one mark per line. Just curious, how many you got four out of four on that one? Same group of good, good, good, good, good, good, though. Number three. I already cheated and wrote the final line in. Hopefully you all clue into that on the test, and that way at least there's no way you can possibly get zero if the final line isn't filled in. Whatever it says show goes there. You may not know the reason, but you'll get something. Che, what am I going to write first? Why? Oh yeah, start out with a, when in doubt, start out with a given. So D, blue Mr. Duke, D, G is parallel to E, F given. Joe, how do I show parallel lines? What symbols do I use? So D, G is parallel to E, F. You know what I now see, Marcus? As I cough after that deep breath. Yeah, you think, wow, I'm busy with that or something. Marcus, what do I see, what letter? Zed, thank you for saying it like a proud Canadian. I don't know, oh, I do know that's what they want me to write next, because what do they have for a reason? That's also a hint that I'm going for. So what's the Zed that I now see? That one there, which means that this angle and this angle are the same. There, I've labeled it, let's write it. I'm going to go angle G, D, F equals angle E, F, D. Long as the D is in the middle of the first one and the F is in the middle of the second one, we're good. And in terms of congruency rules, I'm already pretty sure I'm not going to be using side, side, side, because already right now, Boston, I have angle, which suggests either angle, side, angle, or side, angle, side. Oh, I looked, Mr. Dockett, side, angle, side. Yeah, I noticed that too. Emily, what am I going to write here? What? Really? Yes. Oh, and I'm going to mark it. DG is also the same length as EF. Liam, not only are they parallel, they're same size. Liam, which congruency rule do they want me to use at the very, very bottom there? So I got right now side angle, side. I think that's what, oh, what, Matt, what am I going to write here? First of all, I'm going to write here, because we have to assume the jury's stupid, right? Hello, jury. That's the same in both triangles, same side. By the way, if you want a nice way to label that one, what we used to do back when Geometry is part of Math 12 is we would put a big S on it. Standing four, what do you think, Matt? That was our way of, same side, because if you leave it blank, you might forget that you mentioned it. Or you could just put a single hash mark on it, whatever. And now, Shania, there it is, side, angle, side. Triangle DGF is congruent to triangle FED. One mark per line, half mark, half mark, one mark, one mark, half mark, half mark. If you said equal sides or shared side, the fancy phrase used to be common side. Yeah, I'm looking at you. You're good? How can you not be, right? OK, well, let's keep going. This one, I would consider a nastier one. I don't think I'm going to give you one quite this tough on your test. Liam, can you read to me the first phrase there? Why? How come? Reason. Oh, yeah. OK, so we got that one, Mr. Duc. Here's the problem. What's the new word that we've just used that the jury doesn't understand? There's no midpoint. And that's why you can see the next line, the reason is what, Matt, on the next line? I need to explain to the jury what midpoint means or what I now know because of it. Although you know enough English prefixes, what do you think midpoint would mean? Pardon me? Halfway. In fact, I think what it's saying is, look, if that's the midpoint of that, what can you tell me about this and this? They are the same size. Courtney, that's what they want me to say here. They want me to say, hey, you know what? That means that xp equals yp. Why? Because that's what midpoint means. Jury. Oh, and that gives me side. In fact, I also notice I have a common side here. I'm already thinking side, side, side, or side angles. I'm thinking one with two Ss. Devin, what's this say? What's the next thing that they gave me? Yeah, now you're getting the hang of this little game. Did you say xz equals yz? Yes, and you didn't say xz equals yz, like some people have had, and I've rebuked it. Yes, because we're proud Canadians. Oh, this, since I use two hash marks here, I'll use one hash mark here and here. It's the same as that. Danielle, what do you think I'm going to write here? What's the same side in both triangles? Sorry? Zed's not a side. That's a point. Yep, you know what they want me to write for the stupid jury? Hello, look up, look up, look up, look up. Okay, we've already shown those two are the same size. Those two are the same size. Oh, and by the way, we're gonna use the same side twice. You're gonna write zp equals zp. And I said, I like to show a little s for same size. Same side. Simon, which congruency rule will we force that into here? Oh, and what's gonna go right here? This line, the show. Triangle xpz is congruent to triangle ypz. This one's out of five. One more. Sydney, what am I gonna write first here, kiddo? Turn right. Given. Oh, and I'm gonna mark it on my diagram because to try and keep track of all this in your head, Sarah, ain't happening. So ab, tunk, tunk, bc, tunk, tunk. And I'm already thinking in terms of congruency rules. Side! Boston, what am I gonna write here? Boston, can you read this out loud to me? Say what? Say it again. The jury just, you just lost the jury. And you know which word you lost the jury with? I bet you were gonna have to explain that on the next line. In fact, I'm gonna tell you the next line is going to be definition of bisect. Bisect. Now, what does bisect mean? We have talked about this one a while ago, but we have. Remember what bisect means? It's also a thing of the forward which you should know. Marcus, not just split into two. If I were to, sorry, two equal halves, okay? Because you can cut something into two and have one big chunk and one small chunk. That's not bisect. Bisect, cut exactly in half. Now, you said equal sides, but this says angles. I think what I can conclude is this. Now, let's read. This bisect's angle A, B, C, it cuts this angle in half. Don't write that, don't put a loop there, that's because I'm gonna erase it. I think what that really means is this angle and this angle are the same size. In Sierra, that's what they want me to write for the jury. Angle A, B, D is the same size as angle C, B, D. Is that okay, Courtney? Bisect cuts in half, means I know each half is the same. Now, why is that nice? Ready, Courtney, look up. Side, angle, can I get side angle side? How come? What can you tell me about this side in both triangles? Yes, yes, yes, yes. Can you start to say it? I'm telling you you did. Courtney, what can you tell me about this side in both triangles? What can you tell me about this side in both triangles? Ta-da! We're saying to the jury, hello. I'm using the same side in both triangles. And which congruency rule have I forced that into here, Joe? Which they mentioned, but you know what? Even if they hadn't given it to me, I could have got that. What am I gonna write here? Oh, this line. Triangle bad is congruent to triangle, but could the stop here? So this is the quiz. See if you can figure out number nine. By the way, I'm gonna bet you somewhere in number nine, I'm gonna be using same side because they do share a side. I don't know where, but somewhere. Number 11, oh, Sam, by the way, I'm gonna bet you somewhere in number 11, I'm gonna be using same side because they do share a side. And number 12, and Binder, you know what? I'm gonna bet you somewhere in number 12, I am gonna be using same side because they do share a side. We pause there for a second. So we've done our shuffle. Next unit is trigonometry, the study of triangles, but now from a more mathematical angle perspective. And we start out always when we look at trig with the oldest perhaps theorem about triangles, Pythagoras, which we've talked about already. So trigonometry is the mathematics involved in studying triangles. Triangles been used for years in engineering and architecture and therefore warrant a specific branch of mathematics devoted to them. None of you are in my physics 12. I have a couple of my physics 12s in my other class, but physics 11, we use trig, physics 12, we use trig all the time. Hugely useful, hugely useful. Pythagoras was a famous Greek mathematician who lived in the sixth century. He and his followers, the Pythagoreans, studied many properties of geometric figures. The most famous discovery made by Pythagoras, actually he didn't discover it, he just popularized it so they named it after him, is this relationship. First of all, let's name the sides of a right-angled triangle. Here's a right-angled triangle. The side opposite the 90-degree angle is called the hypotenuse. And traditionally, it's given the symbol lowercase c because it's the third big side and it's the third letter of the alphabet. The other two sides are called legs, not Jordan leg, but leg. And we traditionally use the letters A and B for them. Now actually, I do see some teachers, like Alex teach this way. Leg one plus leg two squared, squared, equals hypotenuse squared. The only reason I don't like it is this is, what number does that lowercase l also look like? Yeah, I've seen some teachers teach it like this. In fact, I think Mr. Raqqa teaches it this way. Side squared plus side squared, my problem is Matt, what number does a lowercase s look like in my atrocious handwriting of five? So I'm yucky all the way around. Did I give him the notes? Did you copy? So I use a squared plus b squared equals c squared, which is easier to remember because it's alphabetical. We can use this to try and find the missing hypotenuse or the missing finding the hypotenuse. By the way, you did do some Pythagoras in some of the geometry packages that we just did too. So this is useful for the test. Find the missing side. First question I asked Sidney is, is this a right angle triangle? Does it have that little box symbol there? It does. Looks 90 degrees. A squared plus b squared equals c squared. Taylor, in our notes, we're gonna write down the Pythagoras equation every time in our homework. If you wanna start going straight to plugging in the numbers, I'm good with that. But we'll do more work in our notes, Courtney, so when you're studying later on, you know what the heck we did. What side a doesn't matter as long as it's not the hypotenuse? What side b doesn't matter as long as it's not the hypotenuse? What side c always the hypotenuse? So I almost always do the hypotenuse first. What sitting where the hypotenuse is? x squared. A squared, I'll go with 56 squared. b squared, I'll go with 33 squared. You could put the 33 there, Boston, and the 56 there. What you could not do, Boston, is not put the x there because the hypotenuse always goes there. How do I get the x squared by itself? It already is! Then I'm gonna crunch the left-hand side. 56 squared plus 33 squared. I get 4225 equals x squared. How to get rid of a squared divide. Matt, the square root of 4225 is gonna be equal to, and you know what the square root of x squared is? Just plain old x. You wanna make sure you know where your square root button is on your calculator. Mine is second functions, that. And this one works out evenly. Most of them won't, but sometimes they will. Equals 65 centimeters. Read example two to me, Boston, my friend. Did you say rectangular? Devin, do you remember what that stands for? Don't. We're gonna draw a little picture. As soon as they give me a word problem, I'm gonna draw a little picture. Little physics acronym I teach my kids is a problem-solving strategy. And because I can't draw a rectangle to save my life, I'm gonna cheat. There's my rectangle. What are the dimensions? 16 by 28. Marcus, I'll be smart enough to put the 16 on the short side and the 28 on the longer side. You don't have to, but don't confuse yourself. And Boston, I think the diagonal is that line right there. That's where I'm gonna put the x. Hey, Boston, how big is this angle right here? How big is this angle right here, folks? Yeah, that's the definition of rectangle. That's why I can use Pythagoras. They don't need to tell me it's a right-angle triangle. Rectangle literally means right angle, but rect is a different word for right in Latin, I think. Am I being asked to find the hypotenuse or a leg? Hypotenuse. So I'll write down a squared plus b squared equals c squared. C squared is x squared. This is gonna be 16 squared plus 28 squared. Find x, I'll do it up here quietly, but see if you can do this on your calculator on your own, too. Am I right? Okay. Sometimes I can ask you to find hypotenuse. Johnson, sometimes I can ask you to find a leg. Oh, I got 102.