 Hey, we're live We looked at Pythagoras and what I said first of all the equation Pythagoras the one that we usually remember Sam because it's alphabetical is a squared plus B squared equals C squared what it really is is Short side plus short side squared squared equals long side squared That gives us a nice shortcut. It means that if I'll probably this is a nice one to give you on a test in the Future as well This means they can give me three sides of a triangle and they can ask me is it a right triangle or not? What they're really saying Natasha is Does Pythagoras work or not in other words? Which of the following side lengths could be used to form a right triangle? The only way that a could be a right triangle is if short side squared plus short side squared Equals long side squared What's the long side here? So here's what we're saying Tanner Does one squared plus three squared Equal five squared and my way of making that a question is I put a question mark above the equal sign That's my way of saying Cheyenne. Are those equal to each other? If it is if they are then that is a right triangle If they're not then I know that that triangle can't have a right angle and Tatiana I didn't have to draw the stupid thing. So it's a nice time saver What is five squared I can do the right hand side in my head? I hope What's one squared? What's three squared? Does one plus nine equal 25 question mark? No, is this a right angle triangle then? Can't possibly be and we just proved it because it doesn't fit Pythagoras one of the reasons Pythagoras is one of the oldest Mathematical formulas out there. It's flexible. You can use it to be clever for all sorts of stuff No, is B a right angle triangle? Well, if it is then short side squared plus short side squared equals long side squared What's the long side in B? Yeah, and I'm going to assume you can find that no matter what order I give you the sides in then you could spot the longest side So Tanner if this is a right angle triangle Seven squared plus eight squared Does that equal nine squared question mark? What's nine squared? What's seven squared? What's eight squared? Does 49 plus 64 equal 81? I don't even think I actually need to do the math because four and six forty and sixty that's a hundred It's already bigger than 81. So, you know what? Not a right angle triangle. Okay, better get perfect on this task. Okay? See if that's a right angle triangle then the short side squared plus the short side squared should equal the long side squared does it Emma? What's the long side in C? Okay, so 25 squared equals question mark. Are they the same? That's my abbreviation for do they equal each other? Seven squared plus 24 squared This one I'm a little more suspicious of because I don't know the answer as to these numbers in my head 25 squared. I know that's 625 Seven squared. I know that's 49, but I do not know what 24 squared is. I know it ends in a six. I think 576 Does 49 plus 576 equals 625? Yeah, you know what that means that means that this is a right angle triangle effect We have a special name for that. We call this a Pythagorean triplet Three numbers that fit Pythagoras. We call them Pythagorean triplets. I'm not gonna ask you to memorize that the nerd within me says Is that okay? Try the last one on your own. Is that a Pythagorean triangle? Is it a right angle triangle or not? And can I find out without drawing it? Tachiana, what's the long side here? So 50 squared is that the same as short side squared plus short side squared? Is it? That's what I'm saying. There I got you to open your calculator I knew I could. 5 times 5 is 25. So 50 times 50 is gonna be 2500. That I can do 3 times 3 is 9. So 30 times 30 is 900. 4 times 4 is 16. 40 times 40 is 1600. What do you think, Nicky? Is that a right angle triangle? It is. And I like this, Nicky, because it's a nice shortcut to try and draw that and prove that and bring out a protractor and measure it. Forget it. I can do it in two lines Okay. Do you all have some room at the bottom of your page right now? Yes? Make it right. You know what? I can type. Actually, Mr. Do it. Click right here. Right? Andy Kint. And I'd like you to draw your best attempt at a circle. I'm gonna cheat because my best attempt at a circle is like Tanner's best attempt at foaming his hair. No, just no. Is that a circle for you guys? Does that look circular on your screen? Okay. On my screen, it doesn't. But that's because the projector is a different resolution. If you ever have a circle in your diagram, if you ever have a circle in your diagram. You know what I'm gonna do today? To help me patrol in Rome. Oh, yeah. That's the advantage of this. Here we go. I can wander around now. Maybe I'll stand back there. Actually, I'm gonna sit right here for now. Draw a line right from the middle to the edge like that. What do you call that line? A line from dead center to the edge. Okay, draw another line like that. What can you tell me about those two lines? In terms of their length. Compared to Tanner. They are the same. Tanner, anytime and starting today, you're gonna notice in the assignment there's gonna be circles. If you ever see a circle and you see a radius, I always say, hey, I'll put double hash marks on or triple hash marks or something to remind myself that they're the same size. Because that often means that you'll have a sauceless triangle or maybe even an equilateral triangle or something like that. So the handy hint is this. If there is a circle, all the plural of radius is radii. So all radii, which is radius, this is are equal. Sometime today, I guarantee in your diagrams, you're gonna see some circles. You're gonna be going, hey, wait a minute. I can't figure this out, and I'm gonna say to you, I bet you if you put hash marks on all the radii, I bet you suddenly there's gonna be an isosceles triangle that you haven't spotted yet. And if it's an isosceles and you know two angles are this and then suddenly it starts to fall apart. Turn the page. So we said this. Turn the page. Quadrilaterals. What's a quadrilateral? It's a four-sided shape. Ashley, what every triangle adds to? 180. You know what every quadrilateral adds to? The sum of the angles in a quadrilateral is 360. And this is kind of a cool proof. I can prove this to you. I can show you why this is. I've drawn a generic, a generic quadrilateral, any four-sided shape. Put a big dot somewhere near the middle. How about right about there? I'm a little off-center. Who cares? And once you've done that, connect that dot to all four corners like this. You can use a ruler if you want to be neat or you can freehand. Pardon me. Technically no. A kite is actually a geometry shape and we named it a kite. But these two would have to be the same length. They'd have to have hash marks. And since they don't, I can't assume they're the same length. But for what it's worth, it is actually a geometry shape. Here's what I want you to notice. By connecting those lines, you've created a bunch of triangles. Yes? How many? I'm going to say this. Every single quadrilateral, if you put a dot in the middle and connect the lines, is four triangles. By the way, what's my abbreviation called for triangles? That's a little triangle symbol. You know what abbreviation for quadrilaterals is? A little four-sided shape. You know what abbreviation for circles is going to be? A little tiny circle. Okay. Nicole, you okay with that? Now, four triangles, that means four of those. If I add up this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, plus this angle, there. Can you go 4 times 180 minus 360 on your calculator? What do you get? What? Every four-sided shape has to add to 360 because in any four-sided shape it really is four triangles but subtract that middle circle because we don't want that in our question. That's a little proof. No numbers. It's one of the few completely visual proofs that was done already about 2,000 years ago. Somebody noticed this and said, hey, I don't need to do numbers. I can just connect the dots. Four-sided shapes. First one is a trapezoid. What's a trapezoid? A trapezoid has two parallel sides. Cole, what's my abbreviation for parallel? How about two parallel lines? Okay, feel free to use that because spelling parallel, you always get the number of L's and R's wrong anyways. A trapezoid has, actually, it's at least two parallel sides. This is a trapezoid here. Okay, what do angle 1 and 2 add to? Why? These two guys form a straight line. What letter is that? This is that interior angles on the same side of the transversal or we also call that co-interior. By the way, Tanner, you're on a roll. What do angle 3 and 4 add to? And what do 1 plus 2 plus 3 plus 4 add to, Tanner? There's your 360. Okay, if you have a trapezoid, Nicky, if you know one angle, you can almost always find the rest. These two add to 180 and those two add to 180. How can I spot a trapezoid? Two parallel sides. There's gonna be some trapezoids in your homework today. Sesame Street is brought today by the trapezoid and turn the page. Should open some windows. Paul, could you open that window there? David, could you open that window there? Can you reach it? Follow your two short. You can reach. Rotate it counterclockwise, up and left. There you go. And then give it a little push. Perfect. Here, Paul, let's do that. Tanner's getting sleepy. We wouldn't want that. Adrian's getting sleepy. Wouldn't want that. Of course, you could always take the hoodie off. Just be thinking. Adrian, my friend, what's the next shape we're looking at right now? Shape B. Can you read this word out to me, please? Parallelogram. The parallelogram has two parallel sides, which actually also means, technically, it's also a trapezoid because if it has two parallel sides, does it have at least one parallel side? All parallelograms are technically trapezoids. This also means that the opposite sides have the same length. So if they tell you it's a parallelogram, you know that this side and this side have the same length and this side and this side have the same length. So I could show that. Let's see. We show parallel with arrows. We show length with hash marks. How about one hash mark on the top and bottom, Nicole? Two hash marks on the left and right. What are these two angles? One and two, add two. Convince me. It's that Iot-sot thing. It's an upside down flip to one. But yeah, it's interior angles on the same side of the transversal or co-interior. Interior angles on the same side of the transversal. They equal 180 degrees. You know what angle is three and four, add two, Tanner? You know what angles one and two, add two. Sorry, you know what angles one and three, add two. Turns out in a parallelogram, any consecutive angles, any angles that are side by side, add to 180. Because, Natasha, there's also a C there. See it? There's also a C there. Upside down and twisted, see it? And there's also a backwards letter C there. All of those add to 180. Now, you ready? Here's the reasoning, Alexis. If one plus two equals 180 and one plus three equals 180, doesn't that mean that those two angles are the same? Yeah. In fact, as it turns out, in a parallelogram, we say that diagonal angles in a parallelogram are equal. That D doesn't look like much of a D, looks like an OE. Let's try that again. Three and two are the same size. One and four are the same size. Speaking of diagonal angles, let's connect the corners on this shape down here with freehand or with the ruler. Connect them. We're getting to your kite, Natasha, but not for a bit yet. If we connect the corners, if you connect the corners of any quadrilateral, we call these diagonals. And as it turns out, if you have a parallelogram, if you have a parallelogram, the diagonals, here's your new word of the day, bisect each other. What does bisect mean? Anybody know? What does bisect mean? More than cross, it means cuts exactly in half. In other words, wash. This length and this length are the same size. This length and this length are the same size. If you know one, you know the other. If you know one, you know the other. So there's our parallelogram. We've done a trapezoid. We've done a parallelogram. Now we're going to look at a special parallelogram. A rectangle. A rectangle. Tanner, a rectangle is a parallelogram, except we had one more quality, Chelsea. All the angles have to be right angles. Nicole, what's this for right angle? That little, this thing here. That says those are right angles. Now because it's a parallelogram, that means all the stuff I just talked about applies. And in a rectangle, the opposite sides are, come here Mr. Dewick. Try that again. Equal to each other. Why is the pen not working? There we go. And misbehaving. Come on Mr. D. The opposite sides. That's supposed to say equal. Okay? You're frustrating me, pen. Let's plug back in and go back with power. Maybe that's the issue. The battery might be getting low. Try this again. Equal. And, Nicole, what's that in the abbreviation for? Yeah. How big is each angle in a rectangle? How big is that angle? 90. And the diagonals bisect each other just like a parallelogram, which means that this one is the same as this one, and this one is the same as this one. Actually, no. It means that all four of these are the same length. This is a little isosceles triangle. This is a little isosceles triangle. This is a little isosceles triangle. This is a little isosceles triangle. And the very last one. And then we're done for today. The rhombus. This is a special type of a parallelogram. It's a parallelogram with four equal sides. Is it the last one? Yes it is. So a rhombus is a parallelogram, which means all the parallelogram stuff applies. But this is the one that has a whole bunch of things. Are you ready? In addition, the diagonals bisect, what did the word bisect mean? Cuts exactly in half. So that means, Nicole, that this length is the same as that length, and this length is the same as that length. Cuts exactly in half. Not only that, the diagonals form a right angle. Tanner, that means that this angle right here is 90 degrees. How big is this angle right here? How big is this angle right here? How big is this angle right here? Do you think Pythagoras might rear its ugly head? Because now there's right angle triangles kicking around all over the place too. And then there's one more. The diagonals bisect the interior, what's that an abbreviation for? Angles of the rhombus. And what does that mean? Once you've written that down, look up. It means that this angle and this angle are the same size. This angle and this angle are the same size. This angle and this angle, those are check marks, are the same size. This angle and this angle are the same size. Really, it means if you know just one of these, suddenly all the rest of them fall apart. Here is your handy reference sheet for quadrilaterals on the back of this page. So when you're doing the homework today, you probably want to keep the back of lesson two out. That has all of the triangle stuff and it has all of the parallel line, bangles, angle stuff. But quadrilaterals, here they all are. And what's your homework? All of geometry package one. But you can skip on page S7. Number seven and eight. And you can skip all of page S13. Now, when I say homework, the rest of this class, I'm going to give you next class to work on it. We're going to mark the take home quiz next class. I'm also going to give you next class to work on your mathematical investigations. That's Monday. When do your mathematical investigations do? Which is when? Next week Friday. So a week and two days from today. Is that all right? Makes sense? Pardon me? Couldn't care less. I'm talking to the whole class right now. Any questions at all? Believe it or not, that's most of the geometry stuff. I got a few more things to show you, but we're going to press pause for a while and let this percolate. Then we're going to start to look at trying to improve things because remember we said that we want to go from inductive reasoning, finding patterns to deductive reasoning, making proofs. What time is it? 1.30. You've got 45 minutes to work on the geometry package or you can work on the take home quiz or you can work on your investigations. I already check lessons one and two last day. If you didn't have lessons one and two done last day and you want to show them to me today, that's all good as well. The remainder of the class is yours. I'm going to go...