Welcome to another webinar. Today we are going to talk about two-dimensional objects. And yes, another friend just entered. All right. So please check the form of question guys. I'm going to give you about one minute. Then we will check the answer together. Remember, I am not asking you to give us answer. Even the first step, couple of step works. Okay. Good luck. Just a reminder for new arrivals, guys. People keep coming. That's nice, man. All right. We are going to talk about two-dimensional geometry today, guys, all of those formulas and stuff. And here is your warm-up question. Please work on the question and share your ideas. Remember, you're not necessarily give us the right answer. That's totally fine. Do your best, okay? okay so far it is one minute guys I know some of our friends just join I can give you 30 more seconds then we are gonna check together remember even you give us the first step what would be the first step you would start that's fine for me please share your ideas whatever you think about question give you 30 more seconds you Alright guys, time is up. Do we have any answers so far? We got one for D and one for B. Okay, so you said one person said it should be B and another person said D. Am I right? Awesome. Alright guys, before I start, can you tell us what do you do as a first step? Meanwhile, I'm going to just show you which parts we need to underline and so forth. We have a given square as you see, which small little squares and is intersected by a line. Okay, we don't know where that line pass through. What is the greatest number? I recommend you to underline those terms. Of a little squares that the line can go through. Okay, you're going to draw a line randomly and you're going to count how many little squares that the line can go through. This is the question basically. Alright. For example, I just don't want to talk to myself. If they give us more answers or ideas, can you share with me please? Alright, that means I believe there is nothing new or I'm going to start. So guys, for example, if you want to use like regular line. It passed through four squares as you see. Okay. If I just use vertical or horizontal line, I am going to have four of those squares as you see. One, two, three, four. It doesn't matter where you start. It's always four, either for horizontal or vertical lines. What if I change it a little bit and make it a little diagonal line? It's not straight but you get the idea. Well, as you see, this line passed through only one. One square basically. Do you think we can get it more? Okay, so hope says nothing that means I will live there is no more ideas guys. What if I Just put the line from here a little bit of it's not really easy to do Little bit from here Well right now with that orange line as you have one two and three little square we pass through okay What else do you think we can get more numbers and how You remember my philosophy guys you can only learn if we just do more interaction If I only explain that to screen well, I'm not sure if you're learning or not Give me something on the chat, please And Type and try to interact with us. What do you think is going on? So I'm doing more answers more ideas, but or now If you can check the chat He says nothing. I believe there is no more answer Please either send to the group or directly so hum guys. I'm not I don't think I can check the chats one by one Because of the timing, you know Oh Someone says you need to go across the entire square Diagonally how so in between two diagonal rows? What do you mean I believe that friend just say like that I I Mean that should give us four to pretty much that but not passing through the corners Or a little bit up or down maybe Yeah, let's try that Diagonally and crossing between squares not through the corners Like like that maybe Okay, look at my straight line, yeah, all right, let's just This is what what they meant. So yes, let's come and try. We have one at the top one two three four Five six seven. Okay, we get seven so far. I see we found both of numbers Friends our friends give us four and seven. Do you think we can make eight? Or is it possible They don't think eight is possible All right, guys. Thanks so much answer was seven exactly whoever says seven I am proud of you. Even you don't say seven. I am so proud of you guys because you try it's about it's a lot Right. No worries The more we try and practice The more we try and practice the more you learn no worries All right If I can Yeah, I see in this week's session we got to talk about two-dimensional geometry those flat surface and their properties So just one more reminder if we have like some of the near friends guys to be able to solve problem Make sure you understand the problem. That's the reason I am telling you Underlying the keyword if it is necessary. It depends on the question though plan how to solve problem That means which method you are going to use are you going to use algebraic ways? Are you going to create equation? Are you going to use table since this topic related geometry? All right, guys, I recommend you to draw a shape Sometimes they're going to give you the whole paragraph all explanation and with no shape without shape It may or may not make so so much sense. I recommend you to the you to do that Get out your plan Of course, like let's say you just work with algebraic base. You just create equation Make sure you are checking your steps and make sure You know, this is the best way look back and check and reflect at the end guys I hope you are checking if Your answer is making sense Or not like if you find the answer I found negative 200 people attended the game It shouldn't be you know case because you cannot say negative 200 people go somewhere. It cannot be negative or you cannot say that's building has 8.5 Stories or whatever? It cannot because it has to be integers. I believe you get the point That's the reason you need to look back and reflect if the answer you found makes sense or not Some of the little properties of the geometry, especially for two-dimensional geometry For all those two-dimensional geometric problems in math kangaroo guys related with shapes or objects That occur in a flat surface flat plane problems can include line segments and rise. Yes These can be parallel or intersect intersect and we are going to talk about a little bit more about these No worries, but kangaroo may ask you to use your knowledge about angles for why these objects as well We have a question about that. No worries a right angle measure Measures 90 degree straight lines measure 180 here as you see this 180 For straight line, this is 90 and if you have complete circle, it's going to be 360 degree guys The sum of angles of any triangle is 180 Well since you are Checking those I have I will ask you a question No worries, because without without the example, it's gonna make no sense We have that type of triangle here guys I will just create it for you and hopefully i'm going to give you whole number as an angle measurement let's say this Part is 6x This part is 2x and this part is x Please just use the information from part number 2 And give me x and all of the other numbers guys. This is your challenge today. Come on your time is start Find x and find all of those interior angles, please it has been 18 seconds you know what does that we have an answer oh we get answer what they say uh x is 20 degrees how they find they didn't say how they found it they said that the other angles were 40 degrees and 120 degrees okay okay guys how do you find that how am i gonna find x equals 20 remember when you solve geometric problems there is a certain steps to solve the problem which method you use 2x plus 6x plus x equals 180 i assume that you use that rule because all of the sum of interlinks is 180 thanks so much how many x do we have here 6 8 9 x is equal to 180 i believe then you solve for x am i right you can say y in the chat box represents yes and for no okay so any question for this one this example problem i just create for you if you type nothing that means i'm gonna assume that there is no question okay we can move on okay all right guys so right triangles have only one 90 degree angle i mean they cannot have more than nine anyway but only one of the angles are right 90 degree anyway the lengths of the sides have special relationship which is a squared plus b squared equals c squared we call them heideggerian theorem and see from that triangle i just want to say that guys we shouldn't necessarily say a squared plus b squared equals c squared because sometimes they messed up with the side lengths and a or b might be hypotenuse you don't know that yet since this is the 90 degree we just show with little box and we call those sides legs guys without to eliminate any misunderstanding i would say leg square it doesn't matter which leg you start plus another leg square gives you hypotenuse square okay this is hypotenuse the longest side anyway this makes more sense okay since if i only explain it doesn't make sense i'm gonna challenge no worries all right let me think about the numbers make more sense if we have that type of triangle i am trying to give you whole number of side lengths by the way this one is 41 this one is 40 and this one is x centimeter i want you to find x please from the part three you just learned it's a little dark okay all right what's x guys come on any answer i guess not yet since i've seen nothing do you get any answer or not so on x equals nine so oh this x is one how i need explanation man come on So line segment AB, here given, and that line D is perpendicular bisector of that line. So this piece is 2x minus 17, and that piece is, let's see, 13. I need you to find x, please, guys. I just said from the given line segment AB, D is a perpendicular bisector here. I just used that from the explanation anyway, and I need you to find x, please, guys. This is 17, by the way. I tried to give you the numbers. Sorry, what was that? x equals 15. Someone says 15. How do they find? Guys, can you type equation you use, and then I can just, you know, solve the rest really quick? Minus 17 equals 13. Okay, so thank you, guys, whoever did that. Our friend says, class, our friend says that since D is a perpendicular bisector, that means if I just say O here, those little line pieces, AO is equal to OB, that means 2x minus 17 has to be equal to 13. Then you got to solve that, you know. First, add 17 both sides, cancel. Then you get 2x is equal to 30, x equals 15, all right? I hope it does make sense. Any question we have about that? I believe no. Okay, so you said nothing, that means I assume there is no question. All right, a little more information about today's topic, perimeters and areas. Perimeter of a 2D dimensional shape is the distance around outside. When you talk about, or when you have given perimeter, guys, you're going to think about outside of the shape, around the shape. The total length of the all segments and curves which make is its outer border, basically. The perimeter of a circle is called circumference. And it's calculated by the formula, 2 pi r. r is the radius of the circle, by the way. Area is the amount of two, of two-dimensional space the figure covers. They say figure covers, guys. Let's say you have some sort of work problem related to geometry, and they ask you to find amount of wrapping paper, let's say, perimeter of a 2D dimensional shape. And they ask you to find amount of wrapping paper you are going to use to cover the, let's say, the desk. In that case, since you cover, guys, you are going to talk about the finding area, or at least service area. Or you have like the box, some sort of rectangular piece of, rectangular prism shape of box. In that case, if you, let's say it's a gift box, if you are going to wrap that gift box some sort of material, you know, you are going to talk about service area. But if you are putting some items in that box, right now we are talking about what, we are going to talk about the three-dimensional geometry. These are just the hints you need to think about when you work with equations. Okay, area of rectangle is basically length times width, it's the basic form. Since triangle is half of the rectangular shape, we can say one half length times width or base times height. Area of circle is pi r squared, guys. I believe you already know that, or you will ask us if you have, if you are confused about those concepts. All right. Number one, this is for you, and I'm going to start time. Please share your ideas with us whenever you find something. All right, it has been one minute. I gave you a little hint. Whichever is really important for the question, I recommend you to underline those. And if there is a shape, I recommend you to use that shape as well. Any ideas, guys? What would be a reasonable answer for that number one? Guys, this is a question from year 2001 and it was the first question. Regularly first questions, at least first couple of questions, are easier ones. I'm just saying, if you are overthinking, please don't. We've got two people and... Oh, we have two people. I cannot see. What did they say? They said rectangle. Okay. At least this is a start. Guys, can you explain how you get rectangle then? I mean, why it's not square but rectangle, for example? anyone how do people get rectangle Since the triangle's legs are 3 and 4, half of its leg will be 1.5 and 2. Both of the legs of the triangle are not equal, so we can take out square and rhombus. Oh, thank you guys. Yeah, one of the friends says, first of all guys, these are perpendicular bisectors, that means I just tried to put the midpoint of CB and AB as you see here, and they said they use perpendicular bisectors, that means these are the 90 degrees, guys. I mean, these two are 90 degrees. We all know that here in the right triangle, this piece is also 90 degrees, and the last one, when you think about those shapes, quadrilateral has to have total 360 degrees, and the last angle is also 90 degrees. Well, we have all of those 90 degrees. We have only two options then, either square or rectangle. Well, you all know that side is, AB is 3 units total, and BC is 4 units, but since these are the midpoints, we have left over the 2 units and 1.5. If sides are not the same, that means we have rectangular shape. If there is any question, please ask me right now, or we are gonna move on. Okay, so I assume there is no question. I'm gonna go number two guys and I'm gonna give you one minute as well. Remember, I don't necessarily want you to find the answer guys, please. Just give us something. weather is kind of dark maybe that's better all right it is 40 seconds by the way guys All right, time's up, class. Do we have anything? One, uh, two people say B. Which one, B or D? 20 centimeters. Okay, 20. All right. What about others? We have more than two people joining here, guys. And also for those two friends, guys, can you give us a little explanation? Why do you think it should be 20 if it is right? How do you find that? You can just type it really quick, please. While I can just underline the keywords. These are rectangle and identical. Okay. Okay, the short side of large. Okay, reference 10. Length of the long side of the large rectangle. So two times the shorter side is the longer side of the large rectangle. No, wait, they miswrote it. All right, they say something that was close actually one two times the shorter side of the small rectangle is the side of the large rectangle okay yes let's just say smaller side or basically width of a small rectangles X since we have all four identical rectangle shapes I can say these sides are also X am I right because look at those little tiny small rectangular shapes and we say width are X well with the same logic if I focus here on the right part with even little tiny small triangle I mean rectangular shape can I say that given lengths are the same lengths are the lengths which are on opposite sides are the same yes we all know that we can say that 2x X plus X 2x is equal to 10 here on the right and left well X should be 5 then if it's 5 centimeter remember this piece little piece also X I forget that since these are all identical objects well and they say what's the length of a longer side length of a longer side of the bigger shape is here guys this one this piece they are asking us to find agency from here to here as you see all right so what we have we have one of the we have two X's here also want 10 units in the middle you are basically adding X plus 10 plus X if you plug in 5 4 X 5 plus 10 plus X this is 10 my way not 16 okay yeah and you should get 4 X because X is 5 we just found at the end guys we are gonna get 15 plus 5 20 is our answer whoever says 20 you get it right thank you guys all right is there any question for this one ask me right now please if you get something look like we have no question otherwise so how would tell me that's anyone am I right how did we get to X oh okay all right guys so it is deal guys I'm just saying that let's just focus that middle tiny rectangular shape of him we all know that we have given length but we don't know the width here so since I don't know that I just get X okay this is the first thing I just thought then they say we use for identical small rectangles can I say that the little tiny small rectangle on the right and left are the same these are kind of look like vertical rectangle shape so these are the same as you see because they said identical identical means they are the same with the same logic can I say this piece is 10 units and that part is also X unit as well you should be right as you see because these are identical well with the same logic guys if I start putting the side lengths of little tiny rectangular shapes in the middle balance again I am saying since they are identical can I say that length is 10 units and width is X units is that true what do you think and here as well guys we get 10 units and X units since they are identical that means they are the same shapes all right is it clear whoever else you asked me or ask our same question or is it it's not clear oops you still have a question about how do we find X guys or no what do you think you kind of try to color code for you so what do they say there have been their problem has been clarified oh okay we can move on that all right I believe so all right thanks guys since you said it's clear then number three is for you guys then I am going to start timer your time starts now I'm trying to be one minute it's gonna be enough All right, it's one minute If there is no answer, I'm gonna give you more time some of you might confuse I understand The one for D Okay, one person says D Can we get more? Because we have a lot of participants not just one participant here. I Expect more guys. Come on Look, you may not find the answer. I don't ask answer. Anyway, just give us what would you do? How would you start or what method would you use? Just hear those Another person says D. Alright, D has two votes so far. What about other answer choices? And a third vote for D. Okay, so far three people says D. A, B, C and E has zero. Let's see what's gonna happen next. Fourth person says D. Alright, we have D as the winner right now, I guess. What do you think? Who do you support? Okay, we get four people for D so far. Others, what do you think? What do you see here? It's raining, that's nice. Others, I hope you are not sleeping in front of the screen because I can see you. No, I cannot actually, I'm messing with you. Person number five says D. Oh, okay. Alright. Many of us says D then. Oops, I would say five, my bad. What am I doing? Five, okay. Alright guys, here is the deal. Since many of you says answer should be D, let's see if it's right. First thing first, it says the diagram is made up of identical small triangles. They are identical and they are small. The first thing I would think, if they're identical, those side lengths should be equal to each other, am I right? Because otherwise, they cannot be identical. This is the first thing you need to think. If these triangles are identical, we need to assume that these are all of the same sides with the same angles. We have all 60 degrees here. Well, if we only add, by the way, these sides are the same as well. If we only add a little line here, let's see what's going to happen. First of all, we have those two lines, those two sides are the same and the angle at the top is 60 degree. If you use the properties of triangles, the bottom you would put X and X, 2X plus 60 give you 180. 180, 2X is equal to 120, X equals 60 guys. As you see, every single angles are 60. That means if I add a little tiny triangle here, by literally connecting those middle, those vertices, that means guys, we are going to get same exact triangle. This is the first thing you need to understand. Because as you see, they say, what is the smallest number of such same triangles can be added? Well, do you think we can extend those lines a little bit like that? Okay, since we just put one line, which is parallel to those lines here, I can add another line like that, which is parallel to those inside lines. With same logic, put other line, please. Okay, assume this is line, it's hard to draw, man. I mean, don't judge me or don't laugh over there in front of yours. It's hard to draw right here. I mean, draw that little, yeah, something like that. Okay, then you can just extend those lines. Look like lines, I know you get the point. And here, and here. It doesn't even connect. That's not nice, man. I need to take some sort of course to have to draw, I guess. Anyway, so let's just see and check those intersection points. With the lines I get, what type of shape do I get here? As you see. Okay, guys, this is what we get. Here is the deal. We get some sort of triangle here in the middle. But if you look at those corners, we get some sort of parallelogram. But we know that those sides are the same, with angles are supposed to be 60 and 60 because total was 180. I can extend those middle lines to make same exact triangle as well. We just keep getting same congruent triangles, guys, here. And I already proved that. Please use the properties of triangles to find angles inside, then you will get all of them are 60 degree, and all of the same side means they are the same exact equilateral triangle. Anyway, so if we use maybe green color, something different, then we can just add them, you know, or count them. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If you say 10, it's not true. 11, 12, it's not 12 either. 13, 14, 15. No, we still have some. 16, 17, 18. Your answer would be 18, guys. I am proud of you. Whoever thinks answer was 18. Any question for this one? This one was a little interesting, guys, because I had to show you how do we find we get the same such exact triangles you are going to be adding. Any question? If you are able to speak here, I mean, the rules were different. I would say, say something. I am giving up on you, but I cannot. So you guys, you are not going to talk. If there is no question, type N, it represents no, please. If there is a question, type your question, so I will explain one more time. People are saying no. All right. Thanks, guys. Let me come over. That's good to hear. Let me go to next. Come on, come on. OK. Another one. And your time just starts. All right guys it has been one minute and ten seconds. Do we have an answer? I'm gonna wait. I mean come on. I have to get some answer or some ideas at least. Or how do we start? Or what was the formula? Okay so four people answered E. Oh okay. E has four votes so far and E is the first place. What about A, B, C or D? Nothing? That's sad. I'm gonna cry for them. All right guys. Let's go. Hey the diameter of the circle keeps getting smaller. Okay thanks so much guys whoever sent that. That's nice. All right so here is the thing guys like regularly how I would start with the question since they said we have given squares with a side length of one I would say you know regularly this since one of the length of a square is also diameter of the circle as you see if I just show you like that. I would say the radius is equal to one over two. But as you see this is the biggest circle which radius has a fraction. Even we are gonna get smaller circles means that radius is going to get smaller and smaller and smaller. I don't like it to be honest. Even they gave us side length of one. Can I change the side length of a square? Yes exactly. Then do that. Look remember guys every single case working with integers even whole numbers integers negative numbers anyway even working with whole numbers is easier compared to working with fractions then play with question make better for you for yourself. Since we are going to find those areas of smaller circles anyway and picture four has the smallest little tiny little circles I would work with picture four first. Let's say these radius of these little tiny smallest circles are one. I can just make my own side. Why not? Because at the end you are gonna compare them that means side is not necessarily case here. It doesn't matter. Okay if I say one that little diameter of the tiny tiny beanie circle is going to be two. Two, four, six and eight. Instead of getting side length of one I can say side length is eight. Okay let's just go with that. In that case if I use eight it's not gonna work. Okay let's just make side length is twelve then. I didn't pay attention here sorry for that. Okay because here the number has to be divided by three and four at the same time. Well we can just use twelve. Sorry for confusion guys. Let's say twelve is gonna work. So when you divide twelve by four it's going to be three but then it's gonna be one and four. One point five. Well we don't like either. Let's make it twenty four. Why not? Okay side length is twenty four. So if it is twenty four guys I am going to say one, two, three, four diameters. When you divide twenty four by four you should get six as a diameter of one of the little tiny triangle and radius is three here. With same logic guys when you divide twenty four by three you should get eight as you know and diameter of little tiny of those circles should be four. Okay here since we have two of those intersect together then you divide twenty four by two you should get twelve and if diameter is twelve radius is six here guys. And this one is easier you can say since diameter is twenty four and radius is twelve. I will just convert those numbers whole number first guys. I mean you don't have to do this but it's easier for me. Also when I observe my kiddos during those 15 plus years. Anyway so how do you find what do we need here guys? We need in which picture is shaded areas grades. We only need shaded area remember. That means we only need to remember area of the circle. They don't ask to non-shade or whatever reason. Don't worry about that. Don't worry about the area of the circle. Only focus area of the circle area of the little circle not square. So if we just start with this one we have pi r square. When you just plug in here r you get pi times three square which is we get nine pi as you see. But we know that guys there is one two three four four times one two three four sixteen of those little tiny circles we get sixteen times nine pi should give you one forty four pi. This is what we get here. What about picture three? You will use the same formula and you can have the same logic here guys. Well what we get here four pi r square. We get pi times four square which is sixteen pi and when you count those one two three four five six seven eight nine. We get nine times sixteen pi. You should get one forty four pi. Interesting. Two of them give us the same area at the total. What about this one? Picture two. We have pi r square pi times six square which gives us thirty six pi as you know. Then we have one two three four of them. Four times thirty six pi gives you one forty four pi. Look at that. We get all three are the same. Look like the last one is gonna be same as that. So we have pi times r square. We have only one of them. Pi times twelve square which is one forty four pi because twelve times twelve is one forty four. That's it. As you see e was the answer and every single of those shaded regions are the same. Any question do we have so far? No question? I'd like something. So go ahead. So there's another way of solving this problem. So you partition this each of the squares into small squares so that each small square contains a circle. Like in picture two you partition the square into a two by two and then you notice that the proportion of area of the circle to area of the square is the same because the circle is inscribed in the square and therefore the since the area of the squares are all the same and the proportion of circle to square is the same then the area of the circles are the same. Exactly. Thanks so much. Okay after these two explanations guys do you have any question about that? Number four. I believe that means no. I'm gonna move on. If you want to use guys side length is one use with fractions or you can go with that. I don't mind I'm just saying I wouldn't do that because I see that many many kiddos make mistakes when they work with fractions. Okay the next question is for you and your time starts. Please give us something not necessarily the answer but at least some sort of way. 25 seconds so far. And 55 seconds. No, it's one minute. Do we have some answer or not yet? One person said D. 65 degrees. And someone also said it is impossible to compute. Okay. We have D versus E. What about others? No more answer? Okay, we got... So the one person who said E switched to D. And then we got two more for E. Wait, wait, wait. So can you tell me? Okay. No one says A, right? No one says A. No one says B. No one says C. We got two for D. Two for D. And two for E. Look at that. That is a tie. Alright. Anyone else? If we get one more person, I'm going to break the tie. Come on. You can guess too. It's totally fine. All right, so I believe no one says D or E again. I'm going to move on. Someone says E, impossible to compute. All right. So far, we get three versus two. That's so exciting, guys. Let's see what's going to be answered. All right. The first thing first, guys, I recommend you to either draw the shape. If the shape is here, put all of the information given under the shape. So they say AD is equal to BC. The first question I will ask you, what does it mean, man? It doesn't make any sense. I mean, what are you going to do? Are you going to transfer that AD here or BC is here? What? I mean, you're going to use the, you know, parallelogram rule, just move that BC here, like solving those Olympic geometry questions. But maybe we are missing something. Remember, well, he said, some of the interior angles in any triangle is 180. If I say X here in the vertex of B, let's just try to find measure of X. Maybe it's going to give us something, you know, just trying to have fun, I guess. So you all know that 50 plus 65 plus X is equal to what? Oh, yes. You are saying 180. I don't want to believe that. 180 is correct answer. So when we add those, 50 plus 65 is 115. You are right. Is 115 or I'm making up? So it's 115, yes. Yeah. Yes, thanks. Okay, 115 plus X is equal to 180. Oh yeah, the next step, you are right. You got to take out 115. Yes, guys. Thanks. You are participating a lot today. I am proud of you guys. So 115, take out this one. X is equal to 65. Okay, let's just put here 65. Okay. And then we can see if that's going to be really important information or just nonsense. Okay, I need you to look at triangle A, C, D and tell me if you figure out something or we can use that 65. Why do I find that 65? I mean, what's the reason behind that? Do we have someone says something so on or not? Why do you find that angle? Is it going to be necessary or not? So we get an isosceles triangle. Yes, guys. I am proud of you. We get isosceles triangle. That means guys, A, D is equal to A, C. Right now, I need you to ignore that little tiny triangle on the right. I just shaded for you. And I need you to look at A, B, C. Do you notice something about A, B, C? Please type in the chat. People are saying that angle A, B, C is congruent to angle B, C, A. That is not true. Yeah, I mean, do you notice something interesting about that triangle A, B, C guys or no? And they say it's isosceles. Exactly. Thanks, at least they get this point. That's nice. So guys, can I say these angles, angle B and angle A here, these are the same. Can I say that? I just show you with the little red color. That means angle C, B, A and angle C, A, B are the same guys. Someone made a little mistake, but it's fine. I still forgive you, no worries. And they're asking us to find A, B, C. A, B, C. We are going to find one of those little red triangles. Well, if I say X here, this one is also gonna be X. Can one of you guys tell me how can I create equation here in the triangle A, B, C to find X please? Look, it's all at a time. Come on, give me the answer. I mean, the equation that I'm gonna solve for you. 2X plus 70 equals 180. I am proud of you guys. 2X plus 70 is equal to 180. What's the first step do I supposed to do? Come on. And then they say 2X is 110. I believe they said take out 70 first, 2X is 110. Then what's next step? And then X equals 55. You divide by two. I believe this is what you meant. X equals 55, guys. All right, any question for this one? I don't hear from you guys. That means there is no question. No questions. All right. Thank you so much, guys. I enjoyed a lot from our today's session. We talked about how to work with two-dimensional shapes. We saw some questions about angles, bisectors, triangles. And so far, these are just some of them I just remember. All right. Also, the first one was interesting, the warm-up question. How many little tiny squares you can pass with the straight line? Even if it was the first question, it was interesting. It was cool. If there is no question, guys, I am going to see you next time. Take care. Bye.

two-dimensional geometry

angles

triangles

perimeter

area

geometric figures

Pythagorean theorem

congruence

parallel lines

problem-solving