 And a little bit of review. I wrote down here, find the following angles. Joe, I'm going to go in order. So what's the first angle I'm going to find here? Absolutely. Let's put a little angle A equals. And I'm either going to use the 20 or the 140, because that's all they've given me. How big is angle A? How do you know it's 140 degrees? You're correct. We called this vertically opposite, and you were looking for x shapes. These two were the same, so 140. I agree. Next angle it wants us to find is angle B. I'm either going to use the 140 or the 20 or angle A. How big is angle B? 40, how do you know? Straight line, I agree. By the way, you'll notice I'm asking more and more, how do you know? We've been talking a bit about proofs. Eventually I'm going to be saying to you, prove it. But right now I'm just saying, how do you know? Angle C, how big? 40, how do you know? You can get it two ways. It forms a straight line with this 140, but we said the real shortcut is if there is an x vertically opposite angles are the same. So it's 40 degrees. How big is angle D? How can I figure that one out? Yeah, Jordan. Triangle, they got to add to 180. And again, I was also, I think you were clever enough to say, well, I'm probably going to use C to find D, because eventually I'm probably going to have to use some stuff I already found. That is the negative with this, is if you get one wrong partway through, all the rest of yours will be wrong. If you use it to find more answers. That's why I'm saying to be careful with your math. It's going to be 180 minus 20 minus angle C is angle D. Jordan, how big? 120? Is that right? And Jordan, that looks OK to me too, because looking at angle D, it looks a bit bigger than 90. Is 120 a bit bigger than 90? I kind of try and glance at these, because they're roughly to scale. In other words, if the angle looked this big Boston, and you tried to say that it was 125 degrees, I'd look at you funny. No, no. Oh, how big is angle E? What does that add to 360? So how big is angle E? 240, how'd you get that? By the way, what did we call this type of an angle? It wasn't obtuse. Do you remember the term? Reflex. I might ask you that one. I don't know. That's a bit more of an obscure one. Try this one on your own here. Again, go alphabetically. I think I'll start with angle F. I'll do it up here. Oh, yeah, I remember this. Is that right? If you're not done, keep going. Look up when you're done and see. That isosceles triangle trick, where you know the two angles are the same. Split the difference by dividing by two. Very helpful to have in your back pocket. Joe, is that OK? Yeah? Yeah, yeah? OK. I kind of mentioned this in passing yesterday at the beginning of this class. Labeling angles. We label angles using three letters, unless it's really, really clear that that's the only angle we can be talking about. For example, if I wanted to talk about angle 1, I couldn't call it angle B, because you wouldn't know whether I was talking about this one or this one or that one. All of those have a vertex at B. So how would I describe that angle right there? I would start far away here, and I would go, oh, it's angle that, that, that. Angle 1 is angle A, B, C. Or, Sam, you could go CBA. But the key is the vertex has to be in the middle, and the two branches have to be on the end. What would you call angle 2? How would you label that one? Matt, how could you label this little guy right here? I don't think you can call it B, because when you say B, it could be that one, or it could be that one, or it could be that one. I need more. You're going to have to use three letters here to label this one. Say that again. B, C, D would be that one. Go in order. So you've got to start here, here, here. That's the one we're trying to talk about. So how would you label it? CBD. CBD. Or you could label it as, first of all, let's make a better D, Mr. Do It. Or, Matt, you could have labeled it as DBC. Either of those is fine. Boston, how would you label angle 3, this little guy right here? No, no, no, you can't start. As soon as you say that again, DBA would be that one. Again, you guys are missing. The order has to be, if you want angle 3, which is on the letter D, that's going to be in the middle of the three letters that you're reading out to me. This one right here, isn't it like this? Yeah? So what letters? What letters? A, D, angle A, D, B. Or you could have gone angle B, D, A. But the D, Boston, has to be in the middle. Yes? OK. Convince me, label angle 4. That's this guy right here. Yes. Shay, how can I label angle 5, this great big one here? See it? Yep. So basically, three letters is enough for us to figure out exactly which angle they're talking about, as long as you can read it properly. So you have to realize, start on the first letter, trace your pencil to the vertex, trace back to the third letter. That's how you know which angle it is. Now, oh, angle 6, angle A, B, D, or DBA. Now, angle 7, because it's the only angle on C, if you said, I'd clue in that that's the one you're talking about, because there is no other angle here. So you can be lazy, Jeremiah, if it's a clear diagram with no extra lines, go ahead. That all right? Make sense? Good. So there's labeling angles. Let's do something with this. Turn the page. Consider the following shape, two lines. And I wrote in the little box here, these two lines look blank. That is, they look like they won't cross. What's the fancy word in math for won't cross? Marcus, these lines look parallel, P-A-R-A-L-L-E-L, which from now on will abbreviate as para. Actually, I'll give you an even better abbreviation. There's a symbol for parallel. I'll show you that in a second. They look parallel. Are they? Well, Danielle, they are, because I did it on my computer. But here's how we show it in geometry. To prove to you that they're parallel, what I would do is I would put two arrows on each one, or three arrows on each one, or one arrow on each one. We use arrows to show that lines are parallel. We use arrows. So if you see this, they look parallel. You can't assume they are. But if you saw that there, do they both have the same number of arrows on them? Then they're parallel. Or you could put two arrows on them, or you could put three arrows, but they have to have the same number. You're going to write the word parallel fairly often. So here's the symbol for parallel, two parallel lines. So if you do that, I'll clue in that that's the word parallel, easier to write and way easier to spell. We're going to take those two parallel lines, Courtney, and we're going to cut across them with a line. A line that cuts across two parallel lines is called a transversal, a transversal. Emily, are these two lines here parallel? Here's the transversal. And if you know that, you can figure out every single other angle if they give you just one. This is 110 degrees. How big is this angle right here? How big is this angle right here? Vertically opposite, or angles on a line. Angles on a supplementary. How big is this angle right here? 70, vertically opposite. How big is this angle right here? What? Shay, you're right. Turns out because these two lines are parallel, you can slide this angle right down into the overlap perfectly. This angle here is 110 degrees. Oh, Shay, how big is this angle right here then? Vertically opposite. How big is this angle right here? Either supplementary or this guy slides down to. How big is this angle right here? In fact, if you have this shape, angle A equals angle D. These two are the same, yes? Angle A also equals angle E. These two are the same, yes? What else is angle A the same as? We'll get to those ones. We're going to get those ones differently. I really want to look at angle C. So angle C is the same as angle B, yes? That looks like an E, Mr. Derrick. Shut up. Angle C is also the same as this one here, angle F. Do you remember I already gave you one letter to look for? What letter are we looking for already? We're going to add a couple of more letters. Once you have parallel lines, we're going to look for capital letter Fs. Don't write this down because we're going to do the C. If you have an F, those two angles are the same. Or it can be a backwards F. Those two angles are the same. We're going to look for Zs. If you have a Z, these two angles are the same. Or if you have a backwards Z, as I'm erasing all my angles foolishly. Or if you have a backwards Z, should have done it on the other diagram, these two angles are the same. And then we'll have one more letter that we're going to look for, a capital letter C. What can you tell me about these two angles inside the letter C? They aren't the same. What do they add to their supplementary? We're going to look at each one of those. And we're going to give that a name and use it as a shortcut. Parallel line rules. These two lines look parallel, but they're not yet until I do this. Now they're parallel. Now they're parallel. If I tell you that this angle right here is 120 degrees, how big is that angle? Sorry, 120. And what we're going to look for from now on is that little pattern there. Except it can be upside down or flipped. So Matt, you said it was 120. I agree with you. What do I mean by upside down or flipped? Well, if you know that this angle right here is 125 degrees, how big is that angle there? Can you see the upside down flipped capital? Those two angles are the same size. And we're going to give that a name. We're going to call that pattern corresponding angles. Corresponding angles. So here's a couple of examples. Which angle am I probably going to find first in this question? Do you think probably going to find A first? And since they only gave me that 120, that's what I'm going to use. Oh, Sesame Street is brought to you by which letter? See it? Upside down, reverse. I'm not going to trace those in anymore because it clutters up the diagrams way too much. But for now, it's there. How big is angle A? Have to be 120. How big is angle B? 60. Why 120? Corresponding. Corresponding. 60, supplementary. Jordan, what am I going to find first in this one? So let's go angle C equals, angle D equals, angle E equals. Let's scroll down a little bit, Mr. Dewick. Angle F equals. Can anybody tell me how big angle C is? I'm either going to use the 124 or the 132. And I'm looking for X's and F's right now. Matt, let's get rid of that. But I agree, 124, correspond, 124, Mr. Dewick. Corresponding. How big is angle D? I'm either going to use the 124 or I'm going to use angle C. How big? 56? How'd you get that? What did we call that two angles that added to 180? It started with an S and rhymed with supplementary. Yes. OK. Now I'm going to move to angle E. I'm either going to use the 132, the 124, or one of these two guys. Ah, backwards letter F. See it? 132 corresponding. How big is angle F? 48? Supplementary. So those are corresponding angles. Yeah. Sorry, I didn't hear you. Do D and F form a straight line? Are you asking if you picked this guy up right here and glued to there, would it form a straight line? You're asking? Do they add to 180? Then they can't, and we've just proved why. And you know what, the part of what we would do in geometry, if you said, are they a straight line? Turns out no. Turn the page. Alternate and interior angles, Zed angles. There's two ways to remember this. I've seen some kids, they call these Zangles. I remember it Zed's the last letter of the alphabet. A is the first letter of the alphabet. I kind of remember they go together, Zed, alternate, interior. You're looking for Zeds. This angle right here and this angle right here are the same size. Because there's a Zed there. Is that okay, Courtney? Maggie, you awake? Give her elbow for me, please. Oh, give her harder elbow for me, please. You awake? You alive? Take your jacket off if you're tired. Okay? Or it can be a backward Zed like this. This angle and this angle are the same size. But Aaron, here's the key. These have to be parallel. These have to be parallel. In other words, your Zed shape can't look like this. It has to look like this. Where the top and the bottom of the Zed are parallel. No crooked Zeds, in other words. But if you know that, these two angles are the same size. Or if it's a backward Zed, these two angles are the same size. Kind of a Zoro thing if you want to bring that into play. Zed angles, alternate interiors, angles, whatever you want to call it. Example, Joe, what am I gonna find first in this one here? Absolutely. How big is angle A? Angle A is 180 degrees. You're telling me that this is a straight line right there? Don't think so. I would say that's probably a bad guess because it doesn't even look what it is. Good. How big angle A is this here? How big? Let's see. See the Zoro. How big, Marcus? I agree. Boston, what angle am I gonna find next? Absolutely. And I'm either gonna use the 47, the 69, or the A. How big is angle B? Can we see it? How do you know? Also a Zed there. You'll start to spot a lot of these better, but I gotta show you them now. And how big is angle C? Oh, what do A, B, and C add to? There's my angles on supplementary. So can somebody go 180 minus that, minus that? How big is angle C have? Oh, or I could even use angles and a triangle. In fact, this is another way to prove that every triangle has to add to 180 degrees. What do you get? 64? Really? Oh yeah, that's right. So we have Xs. We have Fs. Z, z, z, z, z. And one more. We have Xs, vertically opposite. We have Fs corresponding. We have Z angles, Z angles, alternate interior. And then we have interior angles on the same side of the transversal. What? Interior angles on the same side of the transversal. Couldn't they have come up with something shorter for that, Mr. Duke? Well, yeah, they did. These are also called co-interior angles, co-interior. I think there's technically supposed to be a dash right there, co-dash interior, but whatever. My students, we used to actually take this interior angles on the same side of the transversal, and we would abbreviate as interior angles on the same side of the transversal. I ought sought. Which is actually how you pronounce that. So if you say I ought sought or co-interior, or interior angles on the same side of the transversal, you're talking about what I'm about to show you. What letter am I looking for here? What does it say? Now I gotta be fussy. Not a curvy letter C, but a jagged letter C, like that. Angle one and angle two. Sam, see the letter C? They aren't the same size, they don't look the same size. You know what you know, though? Angle one plus angle two adds to 180 degrees. They're supplementary, as it turns out. Or if you have angle three and angle four, a backwards letter C, angle three and angle four add to 180 degrees. Let's look at an example. This one. What angle am I gonna find first? Angle five, so let's write down. Angle five equals, they only gave me two angles. I'm either using the 95 or the 120. Now, Shay, you might be tempted to try doing a zangle, a zed. Is that a proper zed that we're allowed to have? No, because we said the top and bottom have to be parallel, so it's not a zed. I don't think I'm using this 120. I'm staring at this 95. Apparently that's gonna help me find angle five. How? Sorry, I don't think so. Using one we've done earlier, that'd be a straight line if I knew that, but I don't know that. Sesame Street is brought to you today by what letter if you turn it on its side and flip it? Okay, there's an F there. This is what you'll get good at spotting eventually, Jordan, but they can be flipped, they can be sideways, they can be tricky to spot, but that's part of what we're doing, reasoning and looking for patterns. How big is angle five? 95 degrees, because this angle, you know, can slide right over this angle. Mr. Duke, can I do the sound effect? Sure, go ahead. Yeah. What angle am I gonna find next? I'm either gonna use the 95, the 120, or the five. What letter is that sideways and backwards? See, these are the co-interior, the interior angles on the same side of the transversal, the ayaatsats, these two add to 180. So how big is angle six? Have to be. The last one is angle seven. How big is angle seven? Oh, can we see it? Marcus, what? What shape is that? Oh, what's every triangle add to? Do I know that angle? Did I just figure out that angle? Then I know that angle, 180 minus minus. What'd you get, Marcus? I didn't hear. 25? Is he right? I think he's right. This is the page you probably want to keep out in front of you while you're doing the homework. This is the page that eventually most of you will memorize most of these as you get sick and tired of looking them up all the time. This is a summary of everything we've done so far. So you will notice Sydney acute angles, obtuse, complimentary, there's a picture of adding to 90, supplementary, or angles on a line add to 180, a right angle, a straight angle, angles at a point adding to 360, there's the X vertically opposite. Here is a sample of a parallel line and a transversal. Oh, interior angles on the same side of the transversal. We also call this co-interior, or you can abbreviate the first letter of each word. I outside, I'll take that too. Triangles, all triangles add to 180. Isosceles triangle, two sides are equal. And then here's the new one. Equilateral triangle, all three sides are equal. And all three angles are equal. If all three angles are equal, and they add to 180, what does each angle have to be in an equilateral triangle? It's gotta be 60 each. I wrote that there, but it's gotta be 60 of these. Okay, that's a handy reference cheat. Can you open in your workbooks please to page 116? Try number two. Try number three. Try number four. Number five. Now number five is trickier. It says in number five, you'll probably have to find some other blank angles before you can find angle A and angle B and angle C. So for number five, fill in every possible angle that you can as you work your way towards the answer. And then can you look at this great big geometry package that I gave you? We're gonna start to whittle away at this. I think you can do, so where is the notes from today? Mr. Dewick, right there. You can finish up to S6. Now this is not S6, this is not due next class, but I'm saying you're now capable of doing every one of those questions so you can start whittling away at it. It will eventually, Danielle, it sounds like a lot. You're gonna find, I've actually done a full answer key for the whole package. It took me about 45 minutes grand total. You start to get good at it and they start, you get better at spotting the patterns. It starts to get repetitive. Okay? You got about a half an hour, so you can use this and probably have no homework for the weekend. Right click.