 For a lot of these, there may be a second possible answer that works specifically for the reason. So if I don't give the same reason as you, at the very end of the quiz, I'm going to say, does anybody have any different reasons come lawyer with me? Ask me then, because otherwise each question takes like 10 minutes to mark. All right. Number one, did you all say, oh, half mark for the answer, half mark for the reason? Number one, that's got to be 22 degrees, yes, and I would say complimentary. Angle three is going to be 140 degrees, and I would take supplementary or I would take angles on a line. Here, got the quiz, got the quiz? Pause for him. So supplementary or angles on a line. Angle four, 360 minus 90 minus 90 is 180 minus 50. Is it 130? And I used angles at a point, I think, yeah. Half mark for the angle, half mark for the reason. Angle seven, 125 degrees, vertically opposite. Angle eight, 55 degrees, supplementary or angles on a line. Angle nine, 55 degrees, you could go vertically opposite, you could go supplementary, you could go angles on a line, you could go angles at a point, but vertically opposite got you there without having to do any subtraction. I have an answer. So half mark for the degrees, half mark for the reason, mark the half mark wrong, but if you're a lawyer with me afterwards, you might have seen something that I didn't. It's worth lawyering with me on these ones. Here we go, continuing on. Angle C, F, E, C, F, E, oh, that's 90 degrees. Why? I'm going to say supplementary or angles on a line. Angle C, F, D, that's 90, that's 35, 55 degrees. I would take comp or supplementary, complementary or angles on a line. Angle B, F, A, 55 degrees. I used vertically opposite or you could use angles on a line or supplementary. Angle B, F, C, that's going to be 125 degrees. I used vertically opposite, but you could also use angles at a point. Probably some other ones as well, lawyer with me afterwards. Angle B, F, D, oh, that's 180 degrees. I wasn't sure what to write here and then I looked on my yellow handy-dandy reference sheet and I went, oh, straight angle, it's sitting right there, underneath right angle. Lawyer with me afterwards, okay? Next page, Angle 1, 70, corresponding. Angle 2, 70, vertically opposite. Angle 3, 110, supplementary or angles on a line. Question 4, Angle 4, 115, co-interior or interior angles on the same side of the transversal IOTSOT, but co-interior is a heck of a lot easier to write. Angle 5, oh, I see a Z, 115, alt, int, no, because I gave you the yellow sheet so you could actually write down the rule and find it on there. If I was making you memorize the rule, I'd take some slang terms but because I gave you the reference sheet, I'm going to say use the actual terms. Half mark, half mark. Angle 6, well, the only angle they gave me is 70, oh, yeah, 70 corresponding. Little neater, Mr. Dewick, 70 corresponding. I almost didn't see that and I went, oh, yeah, there is a letter F on its side at a 45-degree angle upside down right there, but there is an F. Angle 7, oh, if that's 70, these two add to 180, 110, sub. Angle 8, 70 and I would take either co-interior or corresponding. There may be something else I haven't seen but you've got a lawyer with me afterwards. Angle 1, 50, 50 corresponding. Angle 2, 110 corresponding, those two go together, those two go together. Angle 3, angle 2 and angle 3 are the same size, vertically opposite or you could go Alt-Int because there is a Z. Here, Pen, here, Pen. There is a Z right there too on its side but I saw vertically opposite first. Angle 4, oh, that's 50, that's 110, these add to 180, 120, is that right? Supplementary or angles on a line. By the way, please correct me if I'm doing the arithmetic wrong because I'm doing this on my head. Yes, next page, angle BEH, now angle BEH is that one. Well, the only angle they gave me is 42. Ah, I see a Z because those two are parallel. 42, Alt-Int. Then it says angle EHCEHC, now that's this one here. Oh, 138, go interior or interior angles on the same side of the transversal. Okay, angle GEF, GEF, that's this one here, it's going to be the same size as this one here. 42, vertically opposite or you have an F corresponding would also work. Now we're into triangles, every triangle adds to 180 so angle 1 is going to be 180 minus 85 minus 37. 58 degrees, angle SUM of triangle, SUM of triangle is 180 degrees. Angle 2, 70 degrees because it's an isosceles triangle, angle 3, 70, 70, 140 degrees, angle SUM of triangle. I would also take isosceles triangle but technically I used isosceles to know that those two were the same and then I said adds to 180, that's the angle SUM of triangle rule. Which one isosceles? Yeah, but I'll think bad thoughts about you. What the, oh, it's got to be 60. What's the rule, how come? They're all the same because they're all the same? Oh, that's what we call an equilateral triangle. Oh yeah, there it is on my sheet. Angle 7, 60 degrees vertically opposite. Okay, angle 4, 60 degrees equilateral triangle. Angle 5, oh, zoom, zoom, zoom, I see a Z. So angle 5, 60 degrees, alt int and angle 6. Well, if that's 90 and that's 60, this has got to be 30 degrees. Angle SUM triangles equals 180 degrees. Turn in the page. Angle 5, 40, isosceles triangle. Angle 6, 40 and 40 is 80. 100, angle SUM of triangle. Yes, I take isosceles triangle, but really you use isosceles to find the equal angles and then you said they add to 180, that's the angle SUM of triangle rule. Angle 7, that's 100, this has got to be 80 because they're SUP, elementary. Angle 8, oh, these two are the same length, so if that's 80, that gives me 50 degrees because it's an isosceles triangle and then angle ABC. Oh, that's this great big one here. Oh, one's 40, one's 50, the whole thing is how big? 90. If you wrote, let's try that again, Mr. Duke. If you wrote complimentary, that's fine. If you wrote added them, because that's really what I did. I didn't really use a geometry rule. Well, part of it's 50 and part of it's 40, the whole part's 90. And I guess because they add to 90, you could call it complimentary or lawyer with me afterwards. Angle 1, oh, it's a circle, Mr. Duke, so you should make a note that that and that and that are all the same length because they're all radii. 180 minus 50 is 130 divided by 275? No, 65, yes, yes, I hope. 65 because it's an isosceles triangle, that's 90. 45 because it's an isosceles triangle. Oh, 360 minus 90 minus 50 divided by 2. 110 and I used angles at a point, I think. Oh, is that it? That's it, Mr. Duke. Now, a half mark for each angle, a half mark for each reason, give yourself a lovely score out of 52, don't hand them in just yet because right now some of you need to come lawyer with me, now is the time. So apparently Mr. Duke can't do math, it's out of 42, not 52, is that correct? It's out of 42, yes. Give yourself a score out of 42, now pass them inwards please. Mr. Duke. Lesson five, now I do have an answer key, well I have three answer keys floating around but were there any of these that you were looking at where you were going, Mr. Duke, I have absolutely no idea and I look at the answer key and I'm completely lost. This is your chance to ask. Calling the mind of what, five of ten of these on your test? One or two? Okay, you folks are okay? Yep, you don't have it? Ready? The last lesson, the last lesson. Lesson six, congruency, new term for you guys. We say that two shapes are congruent if they have the same size and the same shape. If two shapes have the same size and the same shape they're congruent. If they have the same shape they're similar. For example, right now on my screen I have a triangle that's a different size from the one behind me on the screen there but it's definitely the same shape because that's how a projector works. A projector makes things larger or smaller but it doesn't distort the shapes otherwise a projector wouldn't work very good. But if two objects, if two shapes have the same size and the same shape we say they're congruent. The symbol for congruency is that's the symbol for same size, that's the symbol for same shape. It's an equal sign with a curly capital N on the top. Same shape, same size, congruent. And what we're going to look at Emma specifically is triangles. We're going to say how do we know that two triangles have the same size and same shape and we're going to try and do it using logic and reasoning again. The following two triangles are congruent. They have the same size and the same shape. And the reason you know they have the same size is that side matches that side. That side, sorry. That side matches that side and that side matches that side. Same size. And Rachel, you also know they have the same shape because that angle matches that angle. That angle matches that angle and that angle matches that angle. We would write this. Triangle A, B, C equals congruent, so an equal sign with a little curly capital N on top. And then once I write out the first triangle, I have to be careful with the order of the second triangle. I have to match up the pairs. A, what does A match up with in the second triangle? What does B match up with in the second triangle? X and what does C match up with? Z. Is that okay, Zach? Zach, I could have done this or triangle. I could have gone B, A, C equals congruent. I won't be fussy on the order of the first one, but the pairs have to match. So if I wrote B first, what would I have to write first in the second one? What goes with B? Sorry, X. What goes with A? What goes with C? Z. So Cole, I'm going to be fussy in the order of the second one. I'm going to make sure you match up the pairs. Usually I do the first one alphabetically, and then I match the second one up, and that way all our answers look the same. It's easier to mark, but as long as you match the pairs up properly, no big deal. And then I wrote, or I could also go B, C, A, and match those up to whatever. When we write a congruency statement then, we have to pair up the matching vertices, the matching angles. So here's the issue that Cole, that we want to ask. If we know three sides and three angles in a triangle, Tanner, if we know three sides and three angles, then we know they're congruent. How little information, Joel, do I need to give you before you can say, stop, that's enough. I can tell you for sure they're the same size, same shape. So now what we're asking is, what's the smallest argument I need to make before it's sit down and shut up, I'm right. Question, what's the minimum amount of info that we need about two triangles before we can definitively say they're congruent, don't give me any more information. There are three main rules, and they're easy to remember. They are on your formula sheet. The first one, next page. The first one we call side, side, side. Except we're not going to write side, side, side. Our abbreviation, write this down next to the title, is going to be SSS. Colleen, if you have two triangles and all three sides match, single hash mark, single hash mark, double hash mark, double hash mark, triple hash mark, triple hash mark, that's enough. You're done. You don't need to tell me that the angles match. I know they match. They are congruent. We could write this. Triangle, A, D, R is congruent to triangle. What does A go with in the second triangle? T, what does D go with in the second triangle? I, and what does R go with in the second triangle? M, reason, not alt int, not vertically opposite reason. So we're going to give you three new reasons today to add to our list. Now these ones you only use for congruencies. And the first one is three sides the same, side, side, side. You ready to left once, wait for a bit, please. We're going to finish the lesson and then go. A second rule. Turns out, Natasha, if you give me two sides that are the same and the angles sandwiched between them, that's enough. Quit. Stop. I don't need any more information. I know the other two sides, the other side and the other two angles match. I know they're congruent. What would be a good abbreviation for side angle side? Do I have a single hash mark? Yes, yes. Do I have a triple hash mark? Yes, yes. Do I have the same size angle between them? Yes. Can't be that angle unfortunately. It's got to be the one between them. That's why we write it as side angle side, not SSA or anything like that. We would write this. Triangle J, K, L is congruent to triangle. Keep going. Now, the trick is, look at the hash marks. So K is on a single hash mark and a blank side. Right? Single hash mark and a blank side. Oh, it's got to be Q. I use the hash mark. Sometimes I can just eyeball it, but often I look at the little hash marks and the symbols Q and then R, reason. Side angle side. So congruency rules, side, side, side, side angle side, there is one more. Angle, side, angle. If you have two triangles and they have two matching angles, double hash angle, double hash angle, dot angle, dot angle, and the side that sandwiched between them, and the side that sandwiched between them is the same. Nikki, that's enough. I don't need any more information. Sorry, I sound like I was chewing you out there. Maybe I was at the same time. No. What would a good abbreviation be for angle, side, angle? ASA. These three are on the back of that sheet in the first column there. I think you'll see side, angle, side, angle, side, angle, and I don't know what order. Probably side, side, side, and then angle, side, angle, and side, angle, side, something like that. There are all three of them right there, yes? We would write this. Triangle, D, E, F is congruent to triangle. Triangle what? Let's see. D goes with H, absolutely. E is on a double hash mark, I, absolutely. And G, reason, angle, side, angle, side, angle. What's not enough? You're going to be given some pictures where they look congruent, but you won't have enough information to prove that they're congruent. Three angles, not enough. Alcoholics Anonymous Association. AAA, American Automobile Association, whatever you want to, three angles is not enough. And then if you have two sides and the angle next to them, except I'm not going to remember it that way, that's how almost every textbook mentions it. There's an easier way to remember this last one and it's one angle and two sides next to them. You can remember badass or whatever you want to remember. That's not congruent. In the time ever, little Simba can say that word in math class. Side, side, side, side, angle, side, angle, side, angle. Let's start applying. Take a look at some of these please. We're going to do some of these together. Put your name on this where it says block. How about block D? Example 1A says write the corresponding congruency. So triangle M and P is equal to triangle is congruent to triangle what? What goes with M? Sorry? K, J, L, Y. What congruency rule do I have here? Side, side, side? No. Which one? That's right down next to here. A, S, A. That's what you're going to do for the rest of number one. Number two simply says are they congruent? If they are, write the congruency rule. If not, just write no. Are these two triangles congruent? Convince me. We got three congruency rules. So what you're looking for is do I have side, side, side? Do I have side, angle, side? Or do I have angle, side, angle? If I don't have one of those three, probably not unless I can add more information. Tanner, do I have two sides and angle between them? Do I have two identical sides and the angle between them? That's enough. I don't need any more information. I know that those two triangles are the same size and same shape. B, here I have angle, side, side. Here I have side, angle, side. They look congruent but I can't prove it. They may or may they not be. And we're talking about, you can't assume they're congruent. We're saying this is how much I need to know to prove beyond a shadow of a doubt that they're congruent. What about C? Why? Okay. So part of your homework right now. Number one. Number two. Next page over. Number three. Let's do three A together. They want you to write yes or no. You can abbreviate that with a Y or an N. Or you could, well, yeah, Y or N. So are these congruent? Yes or no? Sorry? Yes? Convince me. I agree. Side, angle, side. Let's put a big Y and then the congruency statement. How about A, B, C? I can make this a little larger so you guys can see better. Is congruent to, oh, Rob, what goes with A? Yeah, you know, I thought D for a second, but then I went no. A is by a single hash mark. F is by a single hash mark. It's got to be F. What goes with B? Oh, E, that one I can get pretty easily. And that leaves by default D. Now, the diagrams are going to get a bit more complicated. For example, take a look at B. We're talking the top triangle and the bottom triangle. That's what they're asking about. Are those two triangles congruent? Well, I definitely see angle, angle, angle, angle. Do I have angle, side, angle? The answer is this time I do because they're sharing the same side. Doesn't that have to be the same length for both of them? Like if they're sharing the same side, we call it a common side. We have in this triangle angle, side, angle. And in the bottom triangle, angle, shared, side, angle. The answer here is yes, angle, side, angle. Triangle KLM equals congruent to triangle KNM. So, you can't just glance at the diagrams. You're going to have to use some logic and reasoning. For example, take a look at G right here. Side, yes, side. Oh, another side, another side. But I know one more thing that they don't need to tell me that I know because of geometry. Tanner, those two angles are the same. Why? See, which congruency rule is there? It's not obvious, but which congruency is actually there? Side, angle, side is. So, you are allowed to use your geometry rules to fill in missing angles and or sides that you know are the same. This is going to be side, angle, side. It's going to be yes. And, oh heck, let's do the bottom one. Triangle XYZ is congruent to triangle, what goes with X? X does. What goes with Y? Single hash mark V. What goes with Z? W. For A? Oh yeah, it doesn't matter. I generally go alphabetically on the first one and that way all our answers look the same. But, no, yeah, we're fine. By the way, H, side, side, yes, side, side. Tanner, what else do I know? So, I might say that those two are the same. Do I have angle, side, angle, or side, angle, side here? You know what I have? Badass. See it? Angle, side, side. Come on, Pan. Come on, Pan. No, not congruent. It might be. I just can't prove it. I need more information. Okay, so that's what you're doing on this page. You shrink it down. So, part of your homework, you can circle number three. It's right. You know, I will pause up to page S22 for now. We'll start the proofs another day. Now, eventually we're going to be doing all of these. If you want to, you can skip the proofs if you want to get a head start. And then when you get to page, I combined several different handouts here. When you get to page Geometry 10, S15, it starts with the same stuff that we've just been doing. If you want to work through some of those, you can. I do not have answer keys made up yet. I will for next class. First of all, though, if you can, by the end of today, please, hand in the homework from lesson five on proving conjectures. That would be wonderful. I am done. I am shutting up.