Interesting perimeter and area problems

WEBVTT
Kind: captions
Language: en

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Lets do some example problems here, so we have the perimeter

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of each of the outer triangles is 30. So for example if I took

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The sum of this side, this side, and that side I will get 30 and that is

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true of all these outer triangles, these 5 outer triangles.

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They then tell us that the perimeter of FGHIJ

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So FGHIJ the perimeter of this pentagon right over here is 50

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So if I add up that side plus that side plus that side plus that side, plus that side,

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I get 50. And then they say what is the perimeter of the star?

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So the perimeter of the star is really the outsides. if you take the bases

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away of each of these triangles. So it is this side, let me do this in a new color actually

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So the perimeter of the triangle i will do in orange. It is going to be this

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plus that plus that plus that plus that plus that

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i think you get the idea plus that plus that plus that plus that

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So the perimeter of the of the star so let me call this: perimeter perimeter of the star

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it is going to be equal to the perimeter of the 5 triangles

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is equal to perimeter of 5 outer triangles. Just call them 5 triangles like this

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minus their basis, right, if i take the perimeter of all of these sides

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If i added up the part that should not be part of the perimeter of the star

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would be this part,that part, that part,that part, that part and that part.

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those are not the part, those are not the part of the perimeter of the star

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so should be the perimeter of the 5 triangles minus the links of their bases

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links of their 5 bases. So what is the perimeter of the 5 triangles?

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well, the perimeter of each of them is 30, perimeter of 5 of them is going to be 5 times 30

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which is 150, now we want to subtract out the links of their 5 bases

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now the links of their 5 bases if we add them up is the exact perimeter of this in inner pentagon

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right over here. So this inner pentagon has a perimeter 50, that is the sum of the 5 bases.

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So that right over here is 50, so the perimeter of the star is going to be 150 minus 50, or or 100.

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All we need is to get the perimeter of all triangles, subtracted out these bases

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which was the perimeter of the inner pentagon and we are done.

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Now lets do the next problem.

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What is the area of this this quadrilateral, something that has 4 sides of ABCD?

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And this is a little bit we have not seen a figure quite like this just yet,

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it on the right hand side looks like a rectangle, and on the left hand side looks like a triangle

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and this is actually trapezoid, but we can actually as you could imagine

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the way we figure out the area of several triangles splitting it up into pieces we can recognise.

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And the most obvious thing to do here is started A and just drop a rock

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drop an altitude right over here, and so this line right over here is going to hit

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at 90 degrees and we could call this point E.

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And what is interesting here is we can split this up into something we recognize

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a rectangle and a right triangle. But you might say how do, how do we figure out what these you know

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we have this side and that side, so we can figure out the area

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of this rectangle pretty straight forwardly. But how would we, how would we figure out the area of this

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triangle? Well if this side is 6 then that means that this that EC is also going to be 6.

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If AB is 6, notice we have a rectangle right over her, opposite side of a rectangle are equal.

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So if AB equals 6, implies that EC is equal to 6, EC is equal to 6, so EC is equal to 6

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and if EC is equal to 6 then that tells us that DE is going to be 3.

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DE is going to be 3, this distance right over here is going to be 3.

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And we know that because if this is 6, this has to be something that we add to 6 to get 9,

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9 was the length of this entire, of the entire base of this figure right over here.

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9 was this entire distance so 9 minus 6 gives us 3, and now we have all the information that we need

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to figure out the area. The area of this part right over here of this rectangle is

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just going to be 6 times 7, so is going to be equal to 42 plus the area of this triangle right over here.

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Plus the area of this triangle right over here, and that is one half base times height one half.

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The base over her is 3, one half times 3 and the height over here is once again going to be 7

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this is a rectangle, opposite sides are equal, so if this is 7, this is also going to be 7

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one half times 3 times 7, so it is going be 42, lets see.

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3 times 7 is 21, 21 divided by 2 is 10.5, 10.5 so this is going to be equal to 52.5

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52.5 is the area of this entire figure.

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Lets do one more.

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So here I have a bizarre looking, a bizarre looking shape, and we need to figure out its perimeter.

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And it it first seems very daunting because they have only given us this side and this side

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and they have only given us this side right over here.

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And one thing that we are allowed to assume in this and you don't always have to make

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you can't always make that assumption and I just didn't draw it here

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I had time because it would had really crowded out this this diagram.

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Is it all of the angles in this diagrams are right angles,so i could have drawn a right angle here

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a right angle here, a right angle there, right angle there, but as you can see

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it kind of makes things a little bit, it makes things a little bit messy.

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But how do we figure out the perimeter if we don't know these little distances,

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if we don't know these little distances here. And the secret here is to kind of shift the sides

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because all we want to care about is the sum of the sides of the sides.

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So what I will do is a little exercise in shifting the sides. So this side over here I am going to shift

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it and put it right up there, then this side right over here, this length right over here

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I am going to shift and put it right over there. Then let me keep using different colors, and then this

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side right over here I am going to shift it and put it right up here. Then finally Iam going to have

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this side right over here, I can shift it and put it right over there and I think you see

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what is going on right now. Now all of these sides combined are going to be the same as this side

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kind of building, even you know this thing was not a rectangle,its its perimeter is going to be a little bit interesting.

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All we have to think about is this 2 right over here, now lets think about all of these sides that is

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going up and down. So this side i can shift it all the way to the right and go right over here.

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Let me make it clear all inside goes all the way to the end, right that it is the exact same all insde.

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Now this white side I can shift all the way to the right over there, then this green side I can shift

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right over there and then I have, and then I can shift, and then i can shift this.

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Actually let me not shift that green side yet, let me just leave that green side

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so I have not, I have not done anything yet, let me be clear I have not done anything yet with that and

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that I have not shift them over and let me take this side right over here and shift it over.

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So let me take this entire thing and shift over there and shift it over there.

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So before I count these two pieces right over here and we know that each have length 2

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this 90 degrees angle, so this has link to and this has link to.

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Before I count these two pieces, I shifted everything else so I was able to form a rectangle.

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So at least counting everything else I have 7 plus 6, so lets see 7 plus 6

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all of these combined are also going to be 7, plus 7, and all of these characters combined are all also going to be 6,

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plus 6, and then finally I have this 2, right here that I have not counted before, this 2,

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plus this 2, plus this 2. And then we have our perimeter, so what is this giving us,

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7 plus 6 is 13, plus 7 is 20, plus 6 is 26, plus 4 more is equal to 30.

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And we are done.


Triangle area proofs

WEBVTT
Kind: captions
Language: en

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We now know how to find the area of rectangles.

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What I want to do in this video,

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is think about how we can find the areas of triangles.

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So, we're starting here with a right triangle,

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has a 90 degree angle right over here.

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Right triangle ABC.

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And let's think about how we can find this area.

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Well, maybe we can construct a rectangle out of triangle ABC.

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And if we can construct a rectangle out if it, and then, maybe we can

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somehow find our area of part of that rectangle.

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And the best way to construct a rectangle is to really duplicate ABC,

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and then flip it over and put it right on top of this.

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And just to verify, that definitely will be a rectangle.

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So we know that this is 90 degrees right over here.

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Let's say that this is x degrees right over there.

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We have x + 90 + this thing have to be equal to 180.

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So, this thing and this thing have to add up to 90.

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Let's just call this 90-x.

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Now, let's flip this thing and rotate it around,

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so that it will look like this.

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So, then you would have another triangle

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that would look just like this.

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Where now, this right angle,

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on the flipped version is that right angle right over there.

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This angle right over here, this x,

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is now this angle right over there.

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And this 90-x is now this angle, right over there.

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And you can see x + 90-x, that'll give you a right angle.

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And then you have x + 90-x, that gives you a right angle.

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And you have four sides and four right angles,

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you're definitely dealing with a rectangle.

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And this rectangle has two of our original triangles in it.

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So, we can write, we can write that the area of triangle ABC,

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the area of triangle ABC, that's what the brackets mean,

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area of triangle ABC,

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is going to be equal to ? * the area of our entire rectangle.

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And our entire rectangle, let me add another point here.

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Let me call this D, is going to be ? the area of rectangle ABCD.

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And we know how to find the area of rectangle ABCD.

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It's going to be equal to,

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it's going to be equal to the base of the rectangle.

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So, this is going to be equal to ? * and so this part right over here.

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Let me just use a different color.

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The area of ABCD is equal to the base or the width of the rectangle.

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So, that's just the length of BC.

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I'm just putting this in parentheses, it doesn't meanďź I just,

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BC is just the length of this segment right over here

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I'm just putting in the parentheses

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so that we don't get the letters jumbled up.

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So it's going to be this width or this base right here

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times the height of the rectangle.

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So, times AD.

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Timesďź sorry, times AB.

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So, it's this base * this height

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gives us the area of the entire rectangle.

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And the area of a right triangle is half of that.

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So, there we have it.

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It's ? * this base * this height.

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It's the area of a right triangle.

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So, in general if you ever have a right triangle,

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if you ever have a right triangle,

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and this is a right angle right over there which is,

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you need one right angle in order for it to be a right triangle.

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And this base has length B, and this side over here has length H.

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You know that the area,

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the area is going to be equal to ? * the base of the triangle.

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That's the base of the triangle BC * the height of the triangle.

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So you can view it this way, if you look at the actual letter points.

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And if you just use these measures as a base * height,

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it's just ? base * height.

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And we only that this works right now for a right triangle.

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Now, let's think about it for other types of triangles

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that aren't necessarily right triangles.

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So, here I have a kind of an arbitrary triangle ABC.

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And to approach figuring out its area,

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what I wanna do is just split this up into two right triangles.

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So what I'm gonna do is draw a perpendicular from B,

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so I'm just gonna really, if this was, if this was an actual structure,

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you'll just literally drop something straight down from here,

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and that line is going to be perpendicular

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to this base right over here, to AC.

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And let me call that point, let me call that point D.

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And what's useful here is now we've constructed,

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we've turned that one triangle into two right triangles.

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So we can say that the area, let me write it this way,

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we can say the area of triangle ABC,

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that's what we wanna figure out.

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It's equal to the area of this character right here.

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So it's equal to the area of triangle ABD + the area of triangle,

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+ the area of this magenta triangle.

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So, plus the area of BCD, of BCD.

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And this is useful

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because we know how to find the area of right triangles.

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Now, obviously this is 90 degrees

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and this is also going to be 90 degrees.

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The area of ABD is ? base * height.

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So it's going to be ? * the base, which is the length AD.

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So ? * AD, * the height, which is the length of BD right over here,

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assuming that we can figure that out.

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So this * that length, so, BD.

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So that's the area of the blue triangle,

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and I'll find the area of the magenta triangle.

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Well, once again it's a right triangle.

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It's gonna be ?, let me do that in magenta.

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Let me do that in magenta.

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So it's going to be ? * the length of this base right over here,

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which is DC, the length of segment DC * the length of BD again.

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Now, you can factor out a ?, we can factor out a ? BD

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from both of these terms, ? BD.

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So you get ? BD * AD, we're left with AD,

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and that's not the same shade of blue.

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We're left with AD + DC.

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We close the parenthesis.

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And what is AD + DC?

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AD is the length, that length.

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And then DC is this length.

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So if you will add those two lengths, you get the length of AC.

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So this whole thing, this whole thing is the length AC.

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So, we're left with the area of ABC is equal to,

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I'll use in a new color, it's equal to ?,

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and I'll just switch the orders right over here.

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? *, which I'm gonna write in that same color, * AC * BD.

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Now, what is this again?

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Well, this is now ? * the base, which is this AC, * the height,

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which is BD.

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So that's pretty cool.

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It worked for right triangles, and it actually,

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if we know the height of a triangle.

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Now, this isn't one of the sides now.

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For a right triangle, it was one of the sides.

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Now, for this arbitrary triangle, it isn't.

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But if we know it, the area of this triangle

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is still ? * the base * the height.

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Now, what about a triangle like this?

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How can we figure out its area?

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Well, let's try doing the same thing.

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Let's see if we can somehow,

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either construct this out of right triangles,

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or maybe out a right triangle of this

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to make it into another right triangle.

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And the easiest way to do that is,

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we could kind of just drop a rock from right over there.

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And then this, and then where it kind of hits the ground

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would form a right angle.

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And let's call this point D.

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And what we care about, what we wanna find in this,

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what we wanna find the area of,

00:07:29.330 --> 00:07:32.210
we wanna find the area of triangle ABC.

00:07:32.240 --> 00:07:34.100
So that's what we care about.

00:07:34.120 --> 00:07:36.660
Triangle ABC is what we wanna find the area.

00:07:36.660 --> 00:07:40.280
But the area of triangle ABC is gonna be the area

00:07:40.280 --> 00:07:42.650
of this larger right triangle that we've set up.

00:07:42.680 --> 00:07:44.610
That's gonna be this larger triangle.

00:07:53.330 --> 00:07:57.330
Minus the area of this smaller one right over here.

00:07:57.330 --> 00:08:03.830
So, minus the area of ADC, of ADC.

00:08:03.830 --> 00:08:07.140
So I just, this blue one ADB, this is just the whole thing.

00:08:07.140 --> 00:08:11.500
This is the whole thing, just so we're clear of what we're talking about.

00:08:11.500 --> 00:08:13.800
Now, what is the area of ADB.

00:08:13.800 --> 00:08:16.310
Well, we know how to find the area of right triangles.

00:08:16.340 --> 00:08:24.130
ADB, the area of ADB is going to be, is going to be ? * our base,

00:08:24.130 --> 00:08:30.690
which is DB, the length of segment DB * our height,

00:08:30.690 --> 00:08:31.870
which is the length of AD.

00:08:31.920 --> 00:08:34.049
Times AD.

00:08:34.049 --> 00:08:37.480
And then from that we wanna subtract the area of the smaller triangle.

00:08:37.480 --> 00:08:42.299
So that's gonne be ? * our base, which is DC,

00:08:42.299 --> 00:08:46.730
that's the length of our base, DC, * the length of our height,

00:08:46.750 --> 00:08:49.060
which is AD, AD.

00:08:49.100 --> 00:08:53.790
So, here we can factor out a ?, and we can factor an AD.

00:08:53.790 --> 00:08:56.850
A ?, and an AD, so let's do that.

00:08:56.850 --> 00:08:59.330
So we factor out a ? * AD.

00:08:59.330 --> 00:09:05.980
And what's left inside is a DB - a DC.

00:09:11.690 --> 00:09:14.170
Let me put the parenthesis in a white again.

00:09:16.950 --> 00:09:20.380
Well, if you take the length, DB is the length of this whole thing.

00:09:20.380 --> 00:09:25.280
You subtract from that the length DC, you're gonna be left with CB.

00:09:25.280 --> 00:09:31.650
So this character right over here, this is C, this is CB.

00:09:31.650 --> 00:09:38.110
And so the area of ABC is going to be equal to ? * CB,

00:09:38.140 --> 00:09:42.050
I'm just switching the order of multiplication, with a yellow color.

00:09:42.090 --> 00:09:46.220
? * CB * AD

00:09:46.220 --> 00:09:49.140
Now, what is this?

00:09:49.170 --> 00:09:54.560
Once again, it's ? * our base * our height.

00:09:54.590 --> 00:09:56.130
And once again the height in this case,

00:09:56.190 --> 00:10:00.100
'cause it's not a right triangle, it's not one of the sides.

00:10:00.100 --> 00:10:02.230
Someone would have to give you that information.

00:10:02.260 --> 00:10:05.110
You'd have to figure out what this actual height is.

00:10:05.110 --> 00:10:06.700
But what's needed is,

00:10:06.700 --> 00:10:09.820
in any form of triangle the area is ? base * height.

00:10:09.820 --> 00:10:12.870
The height is, kind of, if you have the sides of a right triangle,

00:10:12.890 --> 00:10:14.700
it's easy to figure out.

00:10:14.700 --> 00:10:16.590
For these others, if it's not given,

00:10:16.610 --> 00:10:18.030
you might have to do it in some tricky way,

00:10:18.050 --> 00:10:20.180
figure out the height somehow.

00:10:20.220 --> 00:10:21.850
But anyway, I hope that you find that useful.


Perimeter and area: the basics

WEBVTT
Kind: captions
Language: en

00:00:00.422 --> 00:00:02.880
What I want to do in this video
is a fairly straightforward

00:00:02.880 --> 00:00:07.550
primer on perimeter and area.

00:00:07.550 --> 00:00:09.520
And I'll do perimeter
here on the left,

00:00:09.520 --> 00:00:11.390
and I'll do area
here on the right.

00:00:11.390 --> 00:00:13.723
And you're probably pretty
familiar with these concepts,

00:00:13.723 --> 00:00:16.500
but we'll revisit it
just in case you are not.

00:00:16.500 --> 00:00:19.060
Perimeter is
essentially the distance

00:00:19.060 --> 00:00:20.632
to go around something
or if you were

00:00:20.632 --> 00:00:22.340
to put a fence around
something or if you

00:00:22.340 --> 00:00:26.330
were to measure-- if you were to
put a tape around a figure, how

00:00:26.330 --> 00:00:28.050
long that tape would be.

00:00:28.050 --> 00:00:31.020
So for example, let's
say I have a rectangle.

00:00:33.690 --> 00:00:39.924
And a rectangle is a figure that
has 4 sides and 4 right angles.

00:00:39.924 --> 00:00:41.340
So this is a
rectangle right here.

00:00:41.340 --> 00:00:45.830
I have 1, 2, 3, 4 right angles.

00:00:45.830 --> 00:00:47.800
And it has 4 sides,
and the opposite sides

00:00:47.800 --> 00:00:49.050
are equal in length.

00:00:49.050 --> 00:00:51.700
So that side is going to be
equal in length to that side,

00:00:51.700 --> 00:00:53.660
and that side is equal
in length to that side.

00:00:53.660 --> 00:01:01.370
And maybe I'll label the
points A, B, C, and D.

00:01:01.370 --> 00:01:03.010
And let's say we
know the following.

00:01:03.010 --> 00:01:07.000
And we know that
AB is equal to 7,

00:01:07.000 --> 00:01:12.890
and we know that
BC is equal to 5.

00:01:12.890 --> 00:01:16.500
And we want to know, what
is the perimeter of ABCD?

00:01:16.500 --> 00:01:17.560
So let me write it down.

00:01:17.560 --> 00:01:25.880
The perimeter of
rectangle ABCD is just

00:01:25.880 --> 00:01:28.370
going to be equal to the sum
of the lengths of the sides.

00:01:28.370 --> 00:01:32.490
If I were to build a fence, if
this was like a plot of land,

00:01:32.490 --> 00:01:33.934
I would just have
to measure-- how

00:01:33.934 --> 00:01:35.350
long is this side
right over here?

00:01:35.350 --> 00:01:38.120
Well, we already know
that's 7 in this color.

00:01:38.120 --> 00:01:40.700
So it's that side right
over there is of length 7.

00:01:40.700 --> 00:01:44.350
So it'll be 7 plus
this length over here,

00:01:44.350 --> 00:01:45.820
which is going to be 5.

00:01:45.820 --> 00:01:46.700
They tell us that.

00:01:46.700 --> 00:01:48.280
BC is 5.

00:01:48.280 --> 00:01:49.330
Plus 5.

00:01:49.330 --> 00:01:51.900
Plus DC is going to
be the same length

00:01:51.900 --> 00:01:54.290
is AB, which is
going to be 7 again.

00:01:54.290 --> 00:01:55.670
So plus 7.

00:01:55.670 --> 00:01:58.390
And then finally, DA a or AD,
however you want to call it,

00:01:58.390 --> 00:02:02.290
is going to be the same length
as BC, which is 5 again.

00:02:02.290 --> 00:02:03.790
So plus 5 again.

00:02:03.790 --> 00:02:07.820
So you have 7 plus 5 is 12
plus 7 plus 5 is 12 again.

00:02:07.820 --> 00:02:14.180
So you're going to
have a perimeter of 24.

00:02:14.180 --> 00:02:15.870
And you could go the
other way around.

00:02:15.870 --> 00:02:19.910
Let's say that you
have a square, which

00:02:19.910 --> 00:02:22.070
is a special case
of a rectangle.

00:02:22.070 --> 00:02:27.750
A square has 4 sides and 4 right
angles, and all of the sides

00:02:27.750 --> 00:02:29.260
are equal.

00:02:29.260 --> 00:02:31.135
So let me draw a square here.

00:02:34.470 --> 00:02:36.910
My best attempt.

00:02:36.910 --> 00:02:44.390
So this is A, B, C, D. And
we're going to tell ourselves

00:02:44.390 --> 00:02:46.690
that this right
here is a square.

00:02:46.690 --> 00:02:49.580
And let's say that this
square has a perimeter.

00:02:49.580 --> 00:02:57.310
So square has a perimeter of 36.

00:02:57.310 --> 00:02:59.680
So given that, what is the
length of each of the sides?

00:02:59.680 --> 00:03:01.930
Well, all the sides are going
to have the same length.

00:03:01.930 --> 00:03:03.410
Let's call them x.

00:03:03.410 --> 00:03:09.670
If AB is x, then BC is x,
then DC is x, and AD is x.

00:03:09.670 --> 00:03:11.260
All of the sides are congruent.

00:03:11.260 --> 00:03:12.760
All of these segments
are congruent.

00:03:12.760 --> 00:03:15.170
They all have the same
measure, and we call that x.

00:03:15.170 --> 00:03:17.128
So if we want to figure
out the perimeter here,

00:03:17.128 --> 00:03:20.470
it'll just be x plus x
plus x plus x, or 4x.

00:03:20.470 --> 00:03:25.020
Let me write that. x
plus x plus x plus x,

00:03:25.020 --> 00:03:28.940
which is equal to 4x, which
is going to be equal to 36.

00:03:28.940 --> 00:03:30.670
They gave us that
in the problem.

00:03:30.670 --> 00:03:32.929
And to solve this, 4
times something is 36,

00:03:32.929 --> 00:03:34.720
you could solve that
probably in your head.

00:03:34.720 --> 00:03:37.490
But we could divide
both sides by 4,

00:03:37.490 --> 00:03:39.380
and you get x is equal to 9.

00:03:42.450 --> 00:03:45.220
So this is a 9 by 9 square.

00:03:45.220 --> 00:03:47.210
This width is 9.

00:03:47.210 --> 00:03:50.890
This is 9, and then the height
right over here is also 9.

00:03:50.890 --> 00:03:53.070
So that is perimeter.

00:03:53.070 --> 00:03:56.560
Area is kind of a
measure of how much space

00:03:56.560 --> 00:03:58.420
does this thing take
up in two dimensions?

00:03:58.420 --> 00:04:03.380
And one way to think about area
is if I have a 1-by-1 square,

00:04:03.380 --> 00:04:06.480
so this is a 1-by-1 square--
and when I say 1-by-1,

00:04:06.480 --> 00:04:08.642
it means you only have
to specify two dimensions

00:04:08.642 --> 00:04:10.850
for a square or a rectangle
because the other two are

00:04:10.850 --> 00:04:11.740
going to be the same.

00:04:11.740 --> 00:04:14.860
So for example, you could
call this a 5 by 7 rectangle

00:04:14.860 --> 00:04:17.209
because that immediately
tells you, OK, this side is 5

00:04:17.209 --> 00:04:18.310
and that side is 5.

00:04:18.310 --> 00:04:20.764
This side is 7,
and that side is 7.

00:04:20.764 --> 00:04:22.930
And for a square, you could
say it's a 1-by-1 square

00:04:22.930 --> 00:04:24.596
because that specifies
all of the sides.

00:04:24.596 --> 00:04:26.670
You could really say,
for a square, a square

00:04:26.670 --> 00:04:29.310
where on one side is 1,
then really all the sides

00:04:29.310 --> 00:04:30.060
are going to be 1.

00:04:30.060 --> 00:04:34.080
So this is a 1-by-1 square.

00:04:34.080 --> 00:04:36.750
And so you can view
the area of any figure

00:04:36.750 --> 00:04:41.660
as how many 1-by-1 squares
can you fit on that figure?

00:04:41.660 --> 00:04:45.220
So for example, if we were going
back to this rectangle right

00:04:45.220 --> 00:04:48.710
here, and I wanted to find out
the area of this rectangle--

00:04:48.710 --> 00:04:51.290
and the notation
we can use for area

00:04:51.290 --> 00:04:52.910
is put something in brackets.

00:04:52.910 --> 00:05:00.900
So the area of rectangle
ABCD is equal to the number

00:05:00.900 --> 00:05:04.197
of 1-by-1 squares we can
fit on this rectangle.

00:05:04.197 --> 00:05:05.780
So let's try to do
that just manually.

00:05:05.780 --> 00:05:07.660
I think you already
might get a sense of how

00:05:07.660 --> 00:05:09.430
to do it a little bit quicker.

00:05:09.430 --> 00:05:10.910
But let's put a bunch of 1-by-1.

00:05:10.910 --> 00:05:11.490
So let's see.

00:05:11.490 --> 00:05:15.420
We have 5 1-by-1 squares
this way and 7 this way.

00:05:15.420 --> 00:05:17.540
So I'm going to try my
best to draw it neatly.

00:05:17.540 --> 00:05:26.437
So that's 1, 3, 3,
4, 5, 6, and then 7.

00:05:26.437 --> 00:05:29.490
1, 2, 3, 4, 5, 6, 7.

00:05:29.490 --> 00:05:32.570
So going along one of the
sides, if we just go along

00:05:32.570 --> 00:05:35.440
one of the sides like
this, you could put 7 just

00:05:35.440 --> 00:05:37.500
along one side just like that.

00:05:37.500 --> 00:05:39.190
And then over here,
how many can we see?

00:05:39.190 --> 00:05:40.740
We see that's 1 row.

00:05:40.740 --> 00:05:43.880
And that's 2 rows.

00:05:43.880 --> 00:05:49.520
Then we have 3 rows and
then 4 rows and then 5 rows.

00:05:49.520 --> 00:05:51.140
1, 2, 3, 4, 5.

00:05:51.140 --> 00:05:54.670
And that makes sense because
this is 1, 1, 1, 1, 1.

00:05:54.670 --> 00:05:55.930
Should add up to 5.

00:05:55.930 --> 00:05:59.120
These are 1, 1, 1, 1, 1, 1, 1.

00:05:59.120 --> 00:06:00.550
Should add up to 7.

00:06:00.550 --> 00:06:01.700
Yup, there's 7.

00:06:01.700 --> 00:06:03.530
So this is 5 by 7.

00:06:03.530 --> 00:06:05.230
And then you could
actually count these,

00:06:05.230 --> 00:06:07.525
and this is kind of straight
forward multiplication.

00:06:07.525 --> 00:06:09.650
If you want to know the
total number of cubes here,

00:06:09.650 --> 00:06:12.240
you could count it, or you can
say, well, I've got 5 rows,

00:06:12.240 --> 00:06:13.170
7 columns.

00:06:13.170 --> 00:06:16.870
I'm going to have 35--
did I say cube-- squares.

00:06:16.870 --> 00:06:19.810
I have 5 squares in this
direction, 7 in this direction.

00:06:19.810 --> 00:06:22.130
So I'm going to have
35 total squares.

00:06:22.130 --> 00:06:27.062
So the area of this figure
right over here is 35.

00:06:27.062 --> 00:06:28.770
And so the general
method, you could just

00:06:28.770 --> 00:06:31.061
say, well, I'm just going to
take one of the dimensions

00:06:31.061 --> 00:06:32.940
and multiply it by
the other dimension.

00:06:32.940 --> 00:06:38.140
So if I have a
rectangle, let's say

00:06:38.140 --> 00:06:44.300
the rectangle is
1/2 by 1/2 by 2.

00:06:44.300 --> 00:06:45.516
Those are its dimensions.

00:06:45.516 --> 00:06:46.890
Well, you could
just multiply it.

00:06:46.890 --> 00:06:48.310
You say 1/2 times 2.

00:06:48.310 --> 00:06:50.137
The area here is going to be 1.

00:06:50.137 --> 00:06:51.970
And you might say, well,
what does 1/2 mean?

00:06:51.970 --> 00:06:53.880
Well, it means,
in this dimension,

00:06:53.880 --> 00:06:59.406
I could only fit 1/2
of a 1-by-1 square.

00:06:59.406 --> 00:07:01.280
So if I wanted to do
the whole 1-by-1 square,

00:07:01.280 --> 00:07:02.501
it's all distorted here.

00:07:02.501 --> 00:07:03.500
It would look like that.

00:07:03.500 --> 00:07:05.200
So I'm only doing half of one.

00:07:05.200 --> 00:07:07.700
I'm doing another half
of one just like that.

00:07:07.700 --> 00:07:10.640
And so when you add this
guy and this guy together,

00:07:10.640 --> 00:07:13.240
you are going to
get a whole one.

00:07:13.240 --> 00:07:15.649
Now what about area of a square?

00:07:15.649 --> 00:07:17.190
Well, a square is
just a special case

00:07:17.190 --> 00:07:19.940
where the length and
the width are the same.

00:07:19.940 --> 00:07:24.690
So if I have a square--
let me draw a square here.

00:07:24.690 --> 00:07:31.732
And let's call that XYZ-- I
don't know, let's make this S.

00:07:31.732 --> 00:07:34.020
And let's say I wanted
to find the area

00:07:34.020 --> 00:07:36.340
and let's say I know
one side over here is 2.

00:07:36.340 --> 00:07:41.465
So XS is equal to 2, and I
want to find the area of XYZS.

00:07:41.465 --> 00:07:42.840
So once again, I
use the brackets

00:07:42.840 --> 00:07:46.701
to specify the area of this
figure, of this polygon right

00:07:46.701 --> 00:07:47.450
here, this square.

00:07:47.450 --> 00:07:48.840
And we know it's a square.

00:07:48.840 --> 00:07:50.680
We know all the sides are equal.

00:07:50.680 --> 00:07:53.010
Well, it's a special
case of a rectangle where

00:07:53.010 --> 00:07:55.120
we would multiply the
length times the width.

00:07:55.120 --> 00:07:56.620
We know that they're
the same thing.

00:07:56.620 --> 00:07:58.600
If this is 2, then
this is going to be 2.

00:07:58.600 --> 00:08:00.664
So you just multiply 2 times 2.

00:08:00.664 --> 00:08:02.080
Or if you want to
think of it, you

00:08:02.080 --> 00:08:03.920
square it, which is
where the word comes

00:08:03.920 --> 00:08:05.100
from-- squaring something.

00:08:05.100 --> 00:08:10.156
So you multiply 2 times 2,
which is equal to 2 squared.

00:08:10.156 --> 00:08:11.530
That's where the
word comes from,

00:08:11.530 --> 00:08:15.915
finding the area of a
square, which is equal to 4.

00:08:15.915 --> 00:08:17.540
And you could see
that you could easily

00:08:17.540 --> 00:08:23.999
fit 4 1-by-1 squares
on this 2-by-2 square.


Figuring out all the angles for congruent triangles example

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.660
So we have this larger triangle
here and inside of that,

00:00:03.660 --> 00:00:05.670
we have these other
triangles, and we're

00:00:05.670 --> 00:00:07.480
given this information
right over here.

00:00:07.480 --> 00:00:14.680
That triangle BCD is congruent
to triangle BCA, which

00:00:14.680 --> 00:00:18.270
is congruent to triangle ECD.

00:00:18.270 --> 00:00:20.420
And given just this
information, what

00:00:20.420 --> 00:00:22.130
I want to do in
this drawing, I want

00:00:22.130 --> 00:00:25.380
to figure out what every
angle on this drawing is.

00:00:25.380 --> 00:00:27.700
What's the measure
of every angle?

00:00:27.700 --> 00:00:29.124
So let's see what
we can do here.

00:00:29.124 --> 00:00:30.790
So let's just start
with the information

00:00:30.790 --> 00:00:32.380
that they've actually given us.

00:00:32.380 --> 00:00:40.340
So we know that triangle
BCD is congruent to-- well,

00:00:40.340 --> 00:00:42.264
we know all of these
three triangles

00:00:42.264 --> 00:00:43.430
are congruent to each other.

00:00:43.430 --> 00:00:47.760
So, for example, BCD
is congruent to ECD,

00:00:47.760 --> 00:00:50.320
and so their corresponding
sides and corresponding angles

00:00:50.320 --> 00:00:51.840
will also be congruent.

00:00:51.840 --> 00:00:53.830
So just looking at the
order in which they're

00:00:53.830 --> 00:00:58.270
written B, vertex B
corresponds, in this triangle,

00:00:58.270 --> 00:01:04.410
BCD, corresponds
to vertex B in BCA,

00:01:04.410 --> 00:01:07.610
so this is the B
vertex in BCA, which

00:01:07.610 --> 00:01:11.030
corresponds to the
E vertex in ECD.

00:01:11.030 --> 00:01:13.530
So all-- everything that
I've done in magenta,

00:01:13.530 --> 00:01:16.310
all of these angles
are congruent,

00:01:16.310 --> 00:01:20.900
and then we also know
that the C angle.

00:01:20.900 --> 00:01:26.930
So in BCA-- sorry, BCD,
this angle right over here,

00:01:26.930 --> 00:01:30.706
is congruent to
the C angle in BCA.

00:01:30.706 --> 00:01:33.530
BCA, the C angle
is right over here,

00:01:33.530 --> 00:01:36.430
or C is the vertex
for that angle in BCA.

00:01:36.430 --> 00:01:42.330
And that is also the C angle, I
guess we could call it, in ECD.

00:01:42.330 --> 00:01:46.330
But in ECD, we're talking about
this angle right over here.

00:01:46.330 --> 00:01:48.845
So these three angles are
going to be congruent.

00:01:48.845 --> 00:01:50.620
And I think you could
already guess a way

00:01:50.620 --> 00:01:53.180
to come up with the values
of those three angles.

00:01:53.180 --> 00:01:54.900
But let's keep looking
at everything else

00:01:54.900 --> 00:01:56.170
that they're telling us.

00:01:56.170 --> 00:01:59.250
Finally, we have
vertex D over here.

00:01:59.250 --> 00:02:03.535
So angle-- so this is the last
one in where we listed-- so

00:02:03.535 --> 00:02:08.630
in triangle BCD, this
angle right over here

00:02:08.630 --> 00:02:12.440
corresponds to the A
vertex angle in BCA.

00:02:12.440 --> 00:02:14.390
So BCA, that's
going to correspond

00:02:14.390 --> 00:02:15.812
to this angle right over here.

00:02:15.812 --> 00:02:18.020
It's really the only one
that we haven't labeled yet.

00:02:18.020 --> 00:02:20.450
And that corresponds
to this angle,

00:02:20.450 --> 00:02:22.920
this vertex right over here,
that angle right over there.

00:02:22.920 --> 00:02:24.880
And just to make it
consistent, this C

00:02:24.880 --> 00:02:26.989
should also be
circled in yellow.

00:02:26.989 --> 00:02:28.530
And so we have all
these congruences,

00:02:28.530 --> 00:02:32.030
and now we can come up with some
interesting things about them.

00:02:32.030 --> 00:02:37.520
First of all, here, angle
BCA, angle BCD, and angle DCE,

00:02:37.520 --> 00:02:40.820
they're all congruent, and
when you add them up together,

00:02:40.820 --> 00:02:42.130
you get to 180 degrees.

00:02:42.130 --> 00:02:44.720
If you put them all adjacent,
as they all are right here,

00:02:44.720 --> 00:02:46.260
they end up with
a straight angle,

00:02:46.260 --> 00:02:48.040
if you look at
their outer sides.

00:02:48.040 --> 00:02:49.759
So you have, if
these are each x,

00:02:49.759 --> 00:02:51.300
you have three of
them added together

00:02:51.300 --> 00:02:54.750
have to be 180 degrees, which
tells us that each of these

00:02:54.750 --> 00:02:56.654
have to be 60 degrees.

00:02:56.654 --> 00:02:58.070
That's the only
way you have three

00:02:58.070 --> 00:03:00.820
of the same thing adding
up to 180 degrees.

00:03:00.820 --> 00:03:02.180
Fair enough.

00:03:02.180 --> 00:03:03.820
What else can we do?

00:03:03.820 --> 00:03:06.310
Well, we have these
two characters up here.

00:03:06.310 --> 00:03:09.110
They are both equal and
they add up to 180 degrees.

00:03:09.110 --> 00:03:10.880
They are supplementary,
the only way

00:03:10.880 --> 00:03:12.880
you can have two equal
things that add up to 180

00:03:12.880 --> 00:03:15.140
is if they're both 90 degrees.

00:03:15.140 --> 00:03:17.880
So these two characters
are both 90 degrees.

00:03:17.880 --> 00:03:20.560
Or we could say this is a right
angle, that's a right angle.

00:03:20.560 --> 00:03:22.185
And this is congruent
to both of those,

00:03:22.185 --> 00:03:25.700
so that is also 90
degrees, and then

00:03:25.700 --> 00:03:28.244
we're left with these
magenta parts of the angle.

00:03:28.244 --> 00:03:29.660
And here, we could
just say, well,

00:03:29.660 --> 00:03:33.740
90 plus 60 plus something
is going to add up to 180.

00:03:33.740 --> 00:03:36.090
90 plus 60 is 150.

00:03:36.090 --> 00:03:38.520
So this has to be 30
degrees to add up to 180.

00:03:38.520 --> 00:03:41.280
And if that's 30 degrees,
then this is 30 degrees.

00:03:41.280 --> 00:03:44.744
And then this thing right
over here is 30 degrees.

00:03:44.744 --> 00:03:46.410
And then the last
thing-- we've actually

00:03:46.410 --> 00:03:49.300
done what we said we would do,
we found out all of the angles.

00:03:49.300 --> 00:03:51.404
We could also think
about these outer angles.

00:03:51.404 --> 00:03:52.820
So this-- or not
the outer angles,

00:03:52.820 --> 00:03:54.180
or these combined angles.

00:03:54.180 --> 00:03:59.620
So angle say AC-- or say, angle
ABE, so this whole angle we see

00:03:59.620 --> 00:04:02.030
is 60 degrees.

00:04:02.030 --> 00:04:05.370
This angle is 90 degrees,
and this angle here is 30.

00:04:05.370 --> 00:04:08.770
So what's interesting is
these three smaller triangles,

00:04:08.770 --> 00:04:11.200
they all have the exact
same angles, 30, 60, 90,

00:04:11.200 --> 00:04:14.106
and the exact same side lengths.

00:04:14.106 --> 00:04:15.730
We know that because
they're congruent.

00:04:15.730 --> 00:04:18.313
And what's interesting is when
you put them together this way,

00:04:18.313 --> 00:04:21.320
they construct this larger
triangle, triangle ABE,

00:04:21.320 --> 00:04:22.590
that's clearly not congruent.

00:04:22.590 --> 00:04:23.640
It's a larger triangle.

00:04:23.640 --> 00:04:26.320
It has different
measures for its lengths,

00:04:26.320 --> 00:04:30.250
but it has the same angles
30, 60, and then 90.

00:04:30.250 --> 00:04:32.250
So it's actually similar
to all of the triangles

00:04:32.250 --> 00:04:34.600
that it's made up of.


Example involving an isosceles triangle and parallel lines

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:01.520
So what do we have here?

00:00:01.520 --> 00:00:05.000
We have a triangle, and we
know that the length of AC

00:00:05.000 --> 00:00:06.650
is equal to the length of CB.

00:00:06.650 --> 00:00:08.650
So this is an
isosceles triangle,

00:00:08.650 --> 00:00:11.260
we have two of its legs
are equal to each other.

00:00:11.260 --> 00:00:13.889
And then they also tell us
that this line up here, they

00:00:13.889 --> 00:00:15.180
didn't put another label there.

00:00:15.180 --> 00:00:17.290
Let me put another label
there just for fun.

00:00:17.290 --> 00:00:19.560
Let's call this, you
could even call this a ray

00:00:19.560 --> 00:00:23.900
because it's starting at C,
that line or ray CD is parallel

00:00:23.900 --> 00:00:27.180
to this segment AB over
here, and that's interesting.

00:00:27.180 --> 00:00:29.590
Then they give us
these two angles right

00:00:29.590 --> 00:00:31.040
over here, these
adjacent angles.

00:00:31.040 --> 00:00:32.862
They give it to
us in terms of x.

00:00:32.862 --> 00:00:34.320
And what I want to
do in this video

00:00:34.320 --> 00:00:37.880
is try to figure out what x is.

00:00:37.880 --> 00:00:41.850
And so given that they told us
that this line and this line

00:00:41.850 --> 00:00:44.111
are parallel, and we can
turn this into line CD,

00:00:44.111 --> 00:00:45.860
so it's not just a ray
anymore, so it just

00:00:45.860 --> 00:00:48.570
keeps going on and on
in both directions.

00:00:48.570 --> 00:00:50.610
The fact that they've
given us a parallel line

00:00:50.610 --> 00:00:52.380
tells us that maybe
we can use some

00:00:52.380 --> 00:00:55.200
of what we know about
transversals and parallel lines

00:00:55.200 --> 00:00:57.150
to figure out some
of the angles here.

00:00:57.150 --> 00:01:01.030
And you might recognize
that this right over here,

00:01:01.030 --> 00:01:03.900
this line-- let me do
that in a better color.

00:01:03.900 --> 00:01:08.820
You might recognize that line CB
is a transversal for those two

00:01:08.820 --> 00:01:10.740
parallel lines.

00:01:10.740 --> 00:01:13.400
Let's let me draw both of the
parallel lines a little bit

00:01:13.400 --> 00:01:16.330
more so that you can recognize
that as a transversal,

00:01:16.330 --> 00:01:18.560
and then a few things
might jump out.

00:01:18.560 --> 00:01:21.320
You have this x plus
10 right over here,

00:01:21.320 --> 00:01:23.780
and its corresponding
angle is right down here.

00:01:23.780 --> 00:01:25.764
This would also be x plus 10.

00:01:25.764 --> 00:01:27.180
And if this is x
plus 10, then you

00:01:27.180 --> 00:01:28.910
have a vertical
angle right over here

00:01:28.910 --> 00:01:30.860
that would also be x plus 10.

00:01:30.860 --> 00:01:33.610
Or you could say that you have
alternate interior angles that

00:01:33.610 --> 00:01:35.140
would also be congruent.

00:01:35.140 --> 00:01:38.420
Either way, this base angle
is going to be x plus 10.

00:01:38.420 --> 00:01:40.340
Well, it's an
isosceles triangle.

00:01:40.340 --> 00:01:42.580
So your two base angles
are going to be congruent.

00:01:42.580 --> 00:01:46.170
So if this is x plus 10, then
this is going to be x plus 10

00:01:46.170 --> 00:01:47.290
as well.

00:01:47.290 --> 00:01:50.480
And now we have the three
angles of a triangle expressed

00:01:50.480 --> 00:01:53.087
as functions of--
expressed in terms of x.

00:01:53.087 --> 00:01:55.420
So when we take their sum,
they need to be equal to 180,

00:01:55.420 --> 00:01:56.961
and then we can
actually solve for x.

00:01:56.961 --> 00:02:07.740
We get 2x plus x plus
10 plus x plus 10

00:02:07.740 --> 00:02:12.180
is going to be equal
to 180 degrees.

00:02:12.180 --> 00:02:13.670
And then we can add up the x's.

00:02:13.670 --> 00:02:17.150
So we have a 2x there
plus an x plus another x,

00:02:17.150 --> 00:02:19.382
that gives us 4x.

00:02:19.382 --> 00:02:20.810
4 x's.

00:02:20.810 --> 00:02:23.730
And then we have a plus
10 and another plus 10,

00:02:23.730 --> 00:02:28.420
so that gives us a plus
20, is equal to 180.

00:02:28.420 --> 00:02:31.330
And we can subtract 20
from both sides of that,

00:02:31.330 --> 00:02:36.750
and we get 4x is equal to 160.

00:02:36.750 --> 00:02:42.700
Divide both sides by 4, and
we get x is equal to 40.

00:02:42.700 --> 00:02:43.450
And we're done.

00:02:43.450 --> 00:02:45.070
We've figured out what x is,
and then we could actually

00:02:45.070 --> 00:02:46.444
figure out what
these angles are.

00:02:46.444 --> 00:02:49.200
If this is x plus 10
then you have 40 plus 10,

00:02:49.200 --> 00:02:52.140
this right over here is going
to be a 50-degree angle.

00:02:52.140 --> 00:02:55.867
This is 2x, so 2 times 40,
this is an 80-degree angle.

00:02:55.867 --> 00:02:57.700
It doesn't look at it
the way I've drawn it,

00:02:57.700 --> 00:03:00.600
and that's why you should
never assume anything

00:03:00.600 --> 00:03:01.994
based on how a diagram is drawn.

00:03:01.994 --> 00:03:04.410
So this right over here is
going to be an 80-degree angle,

00:03:04.410 --> 00:03:06.680
and then these two base
angles right over here

00:03:06.680 --> 00:03:10.110
are also going to be 50 degrees.

00:03:10.110 --> 00:03:12.040
So you have 50 degrees,
50 degrees, and 80,

00:03:12.040 --> 00:03:15.370
they add up to 180 degrees.


Another isosceles example problem

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.770
The measure of two angles
of an isosceles triangle

00:00:03.770 --> 00:00:09.340
are 3x plus 5 degrees, we'll
say, and x plus 16 degrees.

00:00:09.340 --> 00:00:11.630
Find all possible values of x.

00:00:11.630 --> 00:00:12.740
So let's think about this.

00:00:12.740 --> 00:00:16.420
Let's draw ourselves an
isosceles triangle or two.

00:00:16.420 --> 00:00:21.301
So it's an isosceles triangle,
like that and like that.

00:00:21.301 --> 00:00:23.050
And actually, let me
draw a couple of them

00:00:23.050 --> 00:00:24.290
just because we want
to think about all

00:00:24.290 --> 00:00:25.920
of the different
possibilities here.

00:00:29.400 --> 00:00:32.202
So we know, from what we know
about isosceles triangles,

00:00:32.202 --> 00:00:34.160
that the base angles are
going to be congruent.

00:00:34.160 --> 00:00:36.360
So that angle is going to
be equal to that angle.

00:00:36.360 --> 00:00:38.620
That angle is going to
be equal to that angle.

00:00:38.620 --> 00:00:42.490
And so what could the 3x plus
5 degrees and the x plus 16,

00:00:42.490 --> 00:00:43.980
what could they be measures of?

00:00:43.980 --> 00:00:45.690
Well, maybe this
one right over here

00:00:45.690 --> 00:00:48.980
has a measure of
3x plus 5 degrees.

00:00:48.980 --> 00:00:51.090
And the vertex is the other one.

00:00:51.090 --> 00:00:56.570
So maybe this one up here
is the x plus 16 degrees.

00:00:56.570 --> 00:00:59.530
The other possibility
is that this

00:00:59.530 --> 00:01:02.000
is describing both base
angles, in which case,

00:01:02.000 --> 00:01:03.380
they would be equal.

00:01:03.380 --> 00:01:07.540
So maybe this one is 3x plus
5, and maybe this one over here

00:01:07.540 --> 00:01:08.745
is x plus 16.

00:01:11.480 --> 00:01:13.520
And then the final
possibility-- actually we

00:01:13.520 --> 00:01:15.150
haven't exhausted
all of them-- is

00:01:15.150 --> 00:01:18.080
if we swap these two--
if this one is x plus 16,

00:01:18.080 --> 00:01:19.730
and that one is 3x plus 5.

00:01:19.730 --> 00:01:21.760
So let me draw ourselves
another triangle.

00:01:25.630 --> 00:01:27.440
And obviously swapping
these two aren't

00:01:27.440 --> 00:01:29.430
going to make a
difference because they

00:01:29.430 --> 00:01:30.825
are equal to each other.

00:01:30.825 --> 00:01:32.950
And then we could make that
one equal to 3x plus 5.

00:01:32.950 --> 00:01:34.480
But that's not going to
change anything either

00:01:34.480 --> 00:01:35.980
because they're
equal to each other.

00:01:35.980 --> 00:01:40.160
So the last situation is
where this angle down here

00:01:40.160 --> 00:01:47.990
is x plus 16, and this
angle up here is 3x plus 5.

00:01:47.990 --> 00:01:49.650
This is 3x plus 5.

00:01:49.650 --> 00:01:51.960
So let's just work
through each of these.

00:01:51.960 --> 00:01:55.220
So in this situation, if
this base angle is 3x plus 5,

00:01:55.220 --> 00:01:57.100
so is this base angle.

00:01:57.100 --> 00:01:58.860
And then we know that
all three of these

00:01:58.860 --> 00:02:00.730
are going to have to
add up to 180 degrees.

00:02:00.730 --> 00:02:05.920
So we get 3x plus
5 plus 3x plus 5

00:02:05.920 --> 00:02:14.260
plus x plus 16 is going to
be equal to 180 degrees.

00:02:14.260 --> 00:02:15.420
We have 3x.

00:02:15.420 --> 00:02:16.370
Let's just add up.

00:02:16.370 --> 00:02:19.140
You have 3x plus
3x, which gives you

00:02:19.140 --> 00:02:23.880
6x, plus another x gives you 7x.

00:02:23.880 --> 00:02:27.950
And then you have 5
plus 5, which is 10,

00:02:27.950 --> 00:02:29.600
plus 16 is equal to 26.

00:02:32.600 --> 00:02:36.030
And that is going
to be equal to 180.

00:02:36.030 --> 00:02:39.070
And then we have, let's
see, 180 minus 26.

00:02:39.070 --> 00:02:45.210
If we subtract 26 from both
sides, we get 180 minus 20

00:02:45.210 --> 00:02:52.790
is 160, minus another 6 is 154.

00:02:52.790 --> 00:02:57.460
You have 7x is equal to 154.

00:02:57.460 --> 00:03:01.408
And let's see how many times--
if we divide both sides by 7,

00:03:01.408 --> 00:03:06.800
7 will go into 140 20 times,
and then you have another 14.

00:03:06.800 --> 00:03:08.235
So it looks like it's 22 times.

00:03:08.235 --> 00:03:10.140
So x is equal to 22.

00:03:10.140 --> 00:03:10.990
Is that right?

00:03:10.990 --> 00:03:13.930
20 times 7 is 140.

00:03:13.930 --> 00:03:18.210
140 plus 14 is 154.

00:03:18.210 --> 00:03:24.700
So we have x is equal
to 22 degrees in this

00:03:24.700 --> 00:03:26.140
the first scenario.

00:03:26.140 --> 00:03:29.200
Now let's think about the
second scenario over here.

00:03:29.200 --> 00:03:32.080
Now we have these
two characters are

00:03:32.080 --> 00:03:34.070
going to be equal to each
other because they're

00:03:34.070 --> 00:03:35.500
both the base angles.

00:03:35.500 --> 00:03:41.100
So you have 3x plus 5
is equal to x plus 16.

00:03:41.100 --> 00:03:43.340
Well, you can subtract
x from both sides.

00:03:43.340 --> 00:03:48.450
And so this becomes 2x
plus 5 is equal to 16.

00:03:48.450 --> 00:03:51.070
We can subtract
5 from both sides

00:03:51.070 --> 00:03:54.510
and you get 2x is equal to 11.

00:03:54.510 --> 00:03:58.190
And then you can
divide both sides by 2,

00:03:58.190 --> 00:04:03.330
and you get x is equal to 11/2.

00:04:03.330 --> 00:04:05.830
So that is our second scenario.

00:04:05.830 --> 00:04:08.270
And then we do our third
scenario right over here.

00:04:08.270 --> 00:04:12.370
If this base angle is x plus
16, then this base angle

00:04:12.370 --> 00:04:14.850
right over here is also
going to be x plus 16.

00:04:14.850 --> 00:04:15.954
They are congruent.

00:04:15.954 --> 00:04:17.329
And then we can
do the same thing

00:04:17.329 --> 00:04:19.420
that we did for
the first scenario.

00:04:19.420 --> 00:04:22.670
All of these angles are going to
have to add up to 180 degrees.

00:04:22.670 --> 00:04:28.665
So we have x plus 16 plus
x plus 16 plus 3x plus 5.

00:04:31.437 --> 00:04:33.020
When you add them
all together, you're

00:04:33.020 --> 00:04:36.570
going to get 180 degrees.

00:04:36.570 --> 00:04:38.020
Now let's add up
all the x terms.

00:04:38.020 --> 00:04:42.880
x plus x is 2x plus 3x is 5x.

00:04:42.880 --> 00:04:46.160
So we get 5x.

00:04:46.160 --> 00:04:50.550
And then you have
16 plus 16 is 32.

00:04:50.550 --> 00:04:52.800
32 plus 5 is 37.

00:04:52.800 --> 00:04:56.055
Plus 37 is equal to 180 degrees.

00:04:59.350 --> 00:05:01.710
Subtract 37 from
both sides, and we

00:05:01.710 --> 00:05:08.180
get 5x is equal to
180 minus 30 is 150.

00:05:08.180 --> 00:05:09.850
So that gets us to 143.

00:05:09.850 --> 00:05:11.620
So it's not going
to divide nicely.

00:05:11.620 --> 00:05:16.260
Divide both sides by 5,
you get x is equal 143/5,

00:05:16.260 --> 00:05:18.260
which we can just leave
as an improper fraction.

00:05:18.260 --> 00:05:19.440
You could write it
as a mixed number

00:05:19.440 --> 00:05:21.530
or however else you
might want to write it.

00:05:21.530 --> 00:05:22.460
And we're done.

00:05:22.460 --> 00:05:29.470
These are the three
possible values

00:05:29.470 --> 00:05:33.730
of x, given the information that
they gave us right up there.


Equilateral and isosceles example problems

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.860
Let's do some example
problems using

00:00:02.860 --> 00:00:06.850
our newly acquired knowledge
of isosceles and equilateral

00:00:06.850 --> 00:00:08.029
triangles.

00:00:08.029 --> 00:00:11.510
So over here, I have kind of
a triangle within a triangle.

00:00:11.510 --> 00:00:14.960
And we need to figure out this
orange angle right over here

00:00:14.960 --> 00:00:17.360
and this blue angle
right over here.

00:00:17.360 --> 00:00:20.550
And we know that
side AB or segment

00:00:20.550 --> 00:00:25.090
AB is equal to segment BC,
which is equal to segment CD.

00:00:25.090 --> 00:00:27.130
Or we could also call that DC.

00:00:27.130 --> 00:00:33.250
So first of all, we see that
triangle ABC is isosceles.

00:00:33.250 --> 00:00:35.637
And because it's isosceles,
the two base angles

00:00:35.637 --> 00:00:36.720
are going to be congruent.

00:00:36.720 --> 00:00:37.690
This is one leg.

00:00:37.690 --> 00:00:39.360
This is the other
leg right over there.

00:00:39.360 --> 00:00:42.350
So the two base angles
are going to be congruent.

00:00:42.350 --> 00:00:47.150
So we know that this angle right
over here is also 31 degrees.

00:00:47.150 --> 00:00:49.550
Well, if we know two of
the angles in a triangle,

00:00:49.550 --> 00:00:51.290
we can always figure
out the third angle.

00:00:51.290 --> 00:00:54.250
They have to add
up to 180 degrees.

00:00:54.250 --> 00:01:00.320
So we could say 31 degrees plus
31 degrees plus the measure

00:01:00.320 --> 00:01:06.920
of angle ABC is
equal to 180 degrees.

00:01:06.920 --> 00:01:08.690
You can subtract 62.

00:01:08.690 --> 00:01:10.870
This right here is 62 degrees.

00:01:10.870 --> 00:01:13.000
You subtract 62 from both sides.

00:01:13.000 --> 00:01:17.350
You get the measure of angle
ABC is equal to-- let's see.

00:01:17.350 --> 00:01:19.980
180 minus 60 would be 120.

00:01:19.980 --> 00:01:21.330
You subtract another 2.

00:01:21.330 --> 00:01:23.860
You get 118 degrees.

00:01:23.860 --> 00:01:28.990
So this angle right over
here is 118 degrees.

00:01:28.990 --> 00:01:30.580
Let me just write it like this.

00:01:30.580 --> 00:01:33.090
This is 118 degrees.

00:01:33.090 --> 00:01:36.540
Well, this angle
right over here is

00:01:36.540 --> 00:01:39.420
supplementary to
that 118 degrees.

00:01:39.420 --> 00:01:43.060
So that angle plus 118 is
going to be equal to 180.

00:01:43.060 --> 00:01:44.820
We already know that
that's 62 degrees.

00:01:44.820 --> 00:01:47.060
62 plus 118 is 180.

00:01:47.060 --> 00:01:52.070
So this right over
here is 62 degrees.

00:01:52.070 --> 00:01:57.140
Now, this angle is one of the
base angles for triangle BCD.

00:01:57.140 --> 00:01:59.960
I didn't draw it that way,
but this side and this side

00:01:59.960 --> 00:02:00.760
are congruent.

00:02:00.760 --> 00:02:03.740
BC has the same length as CD.

00:02:03.740 --> 00:02:06.860
Those are the two legs
of an isosceles triangle.

00:02:06.860 --> 00:02:09.310
You can kind of imagine
it was turned upside down.

00:02:09.310 --> 00:02:10.419
This is the vertex.

00:02:10.419 --> 00:02:11.940
This is one base angle.

00:02:11.940 --> 00:02:13.600
This is the other base angle.

00:02:13.600 --> 00:02:15.680
Well, the base angles are
going to be congruent.

00:02:15.680 --> 00:02:19.140
So this is going to be
62 degrees, as well.

00:02:19.140 --> 00:02:21.690
And then finally, if you want
to figure out this blue angle,

00:02:21.690 --> 00:02:24.470
the blue angle plus these
two 62-degree angles

00:02:24.470 --> 00:02:27.610
are going to have to
add up to 180 degrees.

00:02:27.610 --> 00:02:32.560
So you get 62 plus 62
plus the blue angle, which

00:02:32.560 --> 00:02:38.640
is the measure of
angle BCD, is going

00:02:38.640 --> 00:02:41.006
to have to be equal
to 180 degrees.

00:02:41.006 --> 00:02:42.380
These two characters--
let's see.

00:02:42.380 --> 00:02:45.280
62 plus 62 is 124.

00:02:45.280 --> 00:02:47.710
You subtract 124
from both sides.

00:02:47.710 --> 00:02:52.910
You get the measure of angle
BCD is equal to-- let's see.

00:02:52.910 --> 00:02:55.300
If you subtract 120,
you get 60, and then you

00:02:55.300 --> 00:02:56.460
have to subtract another 4.

00:02:56.460 --> 00:02:59.750
So you get 56 degrees.

00:02:59.750 --> 00:03:02.490
So this is equal to 56 degrees.

00:03:02.490 --> 00:03:03.860
And we're done.

00:03:03.860 --> 00:03:06.060
Now, we could do
either of these.

00:03:06.060 --> 00:03:07.660
Let's do this one
right over here.

00:03:07.660 --> 00:03:11.630
So what is the
measure of angle ABE?

00:03:11.630 --> 00:03:14.620
So they haven't even
drawn segment BE here.

00:03:14.620 --> 00:03:17.090
So let me draw that for us.

00:03:17.090 --> 00:03:20.550
And so we have to figure out
the measure of angle ABE.

00:03:20.550 --> 00:03:23.150
So we have a bunch of
congruent segments here.

00:03:23.150 --> 00:03:27.180
And in particular, we see that
triangle ABD, all of its sides

00:03:27.180 --> 00:03:28.070
are equal.

00:03:28.070 --> 00:03:30.970
So it's an equilateral triangle,
which means all of the angles

00:03:30.970 --> 00:03:31.880
are equal.

00:03:31.880 --> 00:03:34.294
And if all of the angles
are equal in a triangle,

00:03:34.294 --> 00:03:35.585
they all have to be 60 degrees.

00:03:38.230 --> 00:03:41.490
So all of these characters
are going to be 60 degrees.

00:03:41.490 --> 00:03:43.610
Well, that's part
of angle ABE, but we

00:03:43.610 --> 00:03:46.784
have to figure out this
other part right over here.

00:03:46.784 --> 00:03:48.700
And to do that, we can
see that we're actually

00:03:48.700 --> 00:03:50.530
dealing with an
isosceles triangle

00:03:50.530 --> 00:03:52.930
kind of tipped over to the left.

00:03:52.930 --> 00:03:55.130
This is the vertex angle.

00:03:55.130 --> 00:03:56.700
This is one base angle.

00:03:56.700 --> 00:03:58.550
This is the other base angle.

00:03:58.550 --> 00:04:00.550
And the vertex angle
right here is 90 degrees.

00:04:00.550 --> 00:04:02.133
And once again, we
know it's isosceles

00:04:02.133 --> 00:04:07.210
because this side, segment
BD, is equal to segment DE.

00:04:07.210 --> 00:04:10.760
And once again, these two angles
plus this angle right over here

00:04:10.760 --> 00:04:14.335
are going to have to
add up to 180 degrees.

00:04:14.335 --> 00:04:15.590
So you call that an x.

00:04:15.590 --> 00:04:17.000
You call that an x.

00:04:17.000 --> 00:04:20.709
You've got x plus x plus 90
is going to be 180 degrees.

00:04:20.709 --> 00:04:23.560
So you get 2x plus-- let
me just write it out.

00:04:23.560 --> 00:04:25.250
Don't want to skip steps here.

00:04:25.250 --> 00:04:28.720
We have x plus x
plus 90 is going

00:04:28.720 --> 00:04:31.570
to be equal to 180 degrees.

00:04:31.570 --> 00:04:36.850
x plus x is the same thing as
2x, plus 90 is equal to 180.

00:04:36.850 --> 00:04:39.000
And then we can subtract
90 from both sides.

00:04:39.000 --> 00:04:40.980
You get 2x is equal to 90.

00:04:40.980 --> 00:04:42.640
Or divide both sides by 2.

00:04:42.640 --> 00:04:47.610
You get x is equal
to 45 degrees.

00:04:47.610 --> 00:04:50.570
And then we're done
because angle ABE

00:04:50.570 --> 00:04:54.729
is going to be equal to the 60
degrees plus the 45 degrees.

00:04:54.729 --> 00:04:56.520
So it's going to be
this whole angle, which

00:04:56.520 --> 00:04:57.820
is what we care about.

00:04:57.820 --> 00:05:04.097
Angle ABE is going to be 60
plus 45, which is 105 degrees.

00:05:04.097 --> 00:05:05.930
And now we have this
last problem over here.

00:05:05.930 --> 00:05:07.750
This one looks a
little bit simpler.

00:05:07.750 --> 00:05:09.190
I have an isosceles triangle.

00:05:09.190 --> 00:05:10.949
This leg is equal to that leg.

00:05:10.949 --> 00:05:11.990
This is the vertex angle.

00:05:11.990 --> 00:05:13.112
I have to figure out B.

00:05:13.112 --> 00:05:15.320
And the trick here is like,
wait, how do I figure out

00:05:15.320 --> 00:05:17.730
one side of a triangle if
I only know one other side?

00:05:17.730 --> 00:05:19.724
Don't I need to know
two other sides?

00:05:19.724 --> 00:05:21.140
And we'll do it
the exact same way

00:05:21.140 --> 00:05:23.630
we just did that second
part of that problem.

00:05:23.630 --> 00:05:26.470
If this is an isosceles
triangle, which we know it is,

00:05:26.470 --> 00:05:29.790
then this angle is going to
be equal to that angle there.

00:05:29.790 --> 00:05:32.270
And so if we call this x,
then this is x as well.

00:05:32.270 --> 00:05:41.190
And we get x plus x plus
36 degrees is equal to 180.

00:05:41.190 --> 00:05:44.550
The two x's, when you
add them up, you get 2x.

00:05:44.550 --> 00:05:47.210
And then-- I won't
skip steps here.

00:05:47.210 --> 00:05:50.310
2x plus 36 is equal to 180.

00:05:50.310 --> 00:05:55.180
Subtract 36 from both
sides, we get 2x--

00:05:55.180 --> 00:05:58.040
that 2 looks a little bit funny.

00:05:58.040 --> 00:06:04.630
We get 2x is equal to--
180 minus 30 is 150.

00:06:04.630 --> 00:06:08.870
And then you want to
subtract another 6 from 150,

00:06:08.870 --> 00:06:11.480
gets us to 144.

00:06:11.480 --> 00:06:12.570
Did I do that right?

00:06:12.570 --> 00:06:16.360
180 minus 30 is 150, yep, 144.

00:06:16.360 --> 00:06:17.700
Divide both sides by 2.

00:06:17.700 --> 00:06:21.350
You get x is equal
to 72 degrees.

00:06:21.350 --> 00:06:23.880
So this is equal to 72 degrees.

00:06:23.880 --> 00:06:26.250
And we are done.


Equilateral triangle sides and angles congruent

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.300
What we've got over here is a
triangle where all three sides

00:00:04.300 --> 00:00:06.740
have the same length,
or all three sides

00:00:06.740 --> 00:00:08.000
are congruent to each other.

00:00:08.000 --> 00:00:11.470
And a triangle like this
we call equilateral.

00:00:11.470 --> 00:00:13.695
This is an equilateral triangle.

00:00:16.700 --> 00:00:20.840
Now what I want to do is prove
that if all three sides are

00:00:20.840 --> 00:00:25.110
the same, then we know
that all three angles are

00:00:25.110 --> 00:00:28.070
going to have the same measure.

00:00:28.070 --> 00:00:29.916
So let's think how
we can do this.

00:00:29.916 --> 00:00:31.290
Well, first of
all, we could just

00:00:31.290 --> 00:00:34.374
look at-- we know that
AB is equal to AC.

00:00:34.374 --> 00:00:36.040
So let's just pretend
that we don't even

00:00:36.040 --> 00:00:39.080
know that this also
happens to be equal to BC.

00:00:39.080 --> 00:00:41.700
And we know for
isosceles triangles,

00:00:41.700 --> 00:00:45.790
if two legs have
the same length,

00:00:45.790 --> 00:00:49.090
then the base angles
have the same length.

00:00:49.090 --> 00:00:50.250
So let's write this down.

00:00:50.250 --> 00:00:56.260
We know that angle ABC is going
to be congruent to angle ACD.

00:00:56.260 --> 00:00:57.630
So let me write this down.

00:00:57.630 --> 00:01:04.120
We know angle ABC is
congruent to angle ACB.

00:01:07.490 --> 00:01:09.690
So maybe this is my
statement right over here.

00:01:12.800 --> 00:01:16.207
And then we have reason.

00:01:16.207 --> 00:01:17.790
And the reason here,
and I'll write it

00:01:17.790 --> 00:01:21.060
in just kind of shorthand, is
that they're base angles of,

00:01:21.060 --> 00:01:22.540
I guess you could
say an isosceles.

00:01:22.540 --> 00:01:24.620
Because we know that this
side is equal to that side.

00:01:24.620 --> 00:01:26.203
And obviously, this
is an equilateral.

00:01:26.203 --> 00:01:27.460
All of the sides are equal.

00:01:27.460 --> 00:01:29.609
But the fact that these
two legs are equal so

00:01:29.609 --> 00:01:30.900
that the base angles are equal.

00:01:30.900 --> 00:01:39.510
So we say two legs equal
imply base angles are

00:01:39.510 --> 00:01:40.515
going to be equal.

00:01:40.515 --> 00:01:43.140
And that just comes from what we
actually did in the last video

00:01:43.140 --> 00:01:45.069
with isosceles triangles.

00:01:45.069 --> 00:01:47.110
But we can also view this
triangle the other way.

00:01:47.110 --> 00:01:50.790
We could also say that
maybe this angle over here

00:01:50.790 --> 00:01:53.800
is the vertex angle, and maybe
these two are the base angles.

00:01:53.800 --> 00:01:55.220
Because then you
have a situation

00:01:55.220 --> 00:01:58.800
where this side and this side
are congruent to each other.

00:01:58.800 --> 00:02:00.450
And then that angle
and that angle

00:02:00.450 --> 00:02:01.680
are going to the base angles.

00:02:01.680 --> 00:02:07.680
So you could say
angle CAB is going

00:02:07.680 --> 00:02:22.310
to be congruent to angle ABC,
really for the same reason.

00:02:22.310 --> 00:02:26.480
We're now looking at different
legs here and different base

00:02:26.480 --> 00:02:26.980
angles.

00:02:26.980 --> 00:02:29.430
This would now be the
base in this example.

00:02:29.430 --> 00:02:32.870
You can imagine turning an
isosceles triangle on its side.

00:02:32.870 --> 00:02:34.362
But it's the exact same logic.

00:02:34.362 --> 00:02:36.070
So let's just review
what I talked about.

00:02:36.070 --> 00:02:37.920
These two sides are
equal, which imply

00:02:37.920 --> 00:02:39.920
these two base angles are equal.

00:02:39.920 --> 00:02:41.890
These two sides
being equal implied

00:02:41.890 --> 00:02:44.090
these two base angles are equal.

00:02:44.090 --> 00:02:48.820
Well, if ABC is congruent to
ACD and is congruent to CAB,

00:02:48.820 --> 00:02:53.620
then all of these angles
are congruent to each other.

00:02:53.620 --> 00:03:02.560
So then we get angle ABC is
congruent to angle ACB, which

00:03:02.560 --> 00:03:04.860
is congruent to angle CAB.

00:03:04.860 --> 00:03:07.415
And that pretty much gives
us all of the angles.

00:03:07.415 --> 00:03:09.040
So if you have an
equilateral triangle,

00:03:09.040 --> 00:03:11.680
it's actually an equiangular
triangle as well.

00:03:11.680 --> 00:03:13.750
All of the angles are
going to be the same.

00:03:13.750 --> 00:03:15.080
And you actually know
what that measure is.

00:03:15.080 --> 00:03:16.913
If you have three things
that are the same--

00:03:16.913 --> 00:03:20.840
so let's call that x, x,
x-- and they add up to 180,

00:03:20.840 --> 00:03:27.660
you get x plus x plus x is equal
to 180, or 3x is equal to 180.

00:03:27.660 --> 00:03:32.880
Divide both sides by 3, you
get x is equal to 60 degrees.

00:03:32.880 --> 00:03:34.870
So in an equilateral
triangle, not only

00:03:34.870 --> 00:03:37.190
are they all the same
angles, but they're all

00:03:37.190 --> 00:03:44.104
equal to exactly-- they're
all 60 degree angles.

00:03:44.104 --> 00:03:46.020
Now let's think about
it the other way around.

00:03:46.020 --> 00:03:48.080
Let's say I have a triangle.

00:03:48.080 --> 00:03:50.960
Let's say we've got
ourselves a triangle where

00:03:50.960 --> 00:03:54.920
all of the angles are the same.

00:03:54.920 --> 00:04:00.010
So let's say that's point
X, point Y, and point Z.

00:04:00.010 --> 00:04:02.180
And we know that all
the angles are the same.

00:04:02.180 --> 00:04:05.100
So we know that this angle
is congruent to this angle

00:04:05.100 --> 00:04:07.150
is congruent to that angle.

00:04:07.150 --> 00:04:11.240
So what we showed in the last
video on isosceles triangles

00:04:11.240 --> 00:04:13.790
is that if two base
angles are the same,

00:04:13.790 --> 00:04:17.459
then the corresponding legs
are also going to be the same.

00:04:17.459 --> 00:04:21.529
So we know, for example,
that YX is congruent to YZ.

00:04:28.990 --> 00:04:34.105
And we know that because the
base angles are congruent.

00:04:39.760 --> 00:04:46.310
Now we also know that
YZ-- so I'll rewrite YZ--

00:04:46.310 --> 00:04:57.237
is congruent to XZ,
by the same argument.

00:04:57.237 --> 00:04:59.320
But here we're dealing
with different base angles.

00:04:59.320 --> 00:05:00.861
So now, once again,
you can view this

00:05:00.861 --> 00:05:03.470
as almost an isosceles
triangle turned on its side.

00:05:03.470 --> 00:05:05.300
This is the vertex
angle right over here.

00:05:05.300 --> 00:05:06.560
These are the two base angles.

00:05:06.560 --> 00:05:07.980
This would be the base now.

00:05:07.980 --> 00:05:11.590
And we know that because these
two base angles are congruent.

00:05:11.590 --> 00:05:13.430
So by the same logic.

00:05:13.430 --> 00:05:16.070
Over in this first
case, the base angles

00:05:16.070 --> 00:05:17.940
were this angle and that angle.

00:05:17.940 --> 00:05:19.930
In the second case,
the base angles

00:05:19.930 --> 00:05:21.490
are that angle and that angle.

00:05:21.490 --> 00:05:23.020
And actually let
me write it down.

00:05:23.020 --> 00:05:25.780
The base angles in
this first case--

00:05:25.780 --> 00:05:27.590
let me do that
same magenta-- are

00:05:27.590 --> 00:05:38.130
angle YXZ is congruent
to angle YZX.

00:05:41.075 --> 00:05:42.200
That was in the first case.

00:05:42.200 --> 00:05:43.480
These are the base angles.

00:05:43.480 --> 00:05:46.070
So based on the proof we
saw in the last video,

00:05:46.070 --> 00:05:48.120
that implies these
sides are congruent.

00:05:48.120 --> 00:05:52.100
Here, we have these
two base angles.

00:05:52.100 --> 00:05:53.730
Let me do that in green.

00:05:53.730 --> 00:06:01.510
Angle XYZ is congruent
to angle YXZ.

00:06:05.410 --> 00:06:08.010
And so that implies that
these two guys right over here

00:06:08.010 --> 00:06:09.020
are congruent.

00:06:09.020 --> 00:06:10.220
Well, there we've proved it.

00:06:10.220 --> 00:06:14.090
We've said that this side
YX is congruent to YZ.

00:06:14.090 --> 00:06:16.190
And we've shown that
YZ is congruent to XZ.

00:06:16.190 --> 00:06:18.750
So all of the sides are
congruent to each other.

00:06:18.750 --> 00:06:20.750
So once again, if you
have all the angles equal,

00:06:20.750 --> 00:06:22.541
and they're going to
have to be 60 degrees,

00:06:22.541 --> 00:06:24.430
then you know that
all of the sides

00:06:24.430 --> 00:06:26.710
are going to be equal as well.

00:06:26.710 --> 00:06:28.456
They're going to be congruent.


Congruent legs and base angles of isosceles triangles

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.089
So we're starting off
with triangle ABC here.

00:00:04.089 --> 00:00:05.880
And we see from the
drawing that we already

00:00:05.880 --> 00:00:09.650
know that the length of AB
is equal to the length of AC,

00:00:09.650 --> 00:00:14.700
or line segment AB is
congruent to line segment AC.

00:00:14.700 --> 00:00:18.530
And since this is a triangle
and two sides of this triangle

00:00:18.530 --> 00:00:21.010
are congruent, or they
have the same length,

00:00:21.010 --> 00:00:23.490
we can say that this is
an isosceles triangle.

00:00:26.960 --> 00:00:30.500
Isosceles triangle, one of the
hardest words for me to spell.

00:00:30.500 --> 00:00:31.852
I think I got it right.

00:00:31.852 --> 00:00:33.560
And that just means
that two of the sides

00:00:33.560 --> 00:00:35.290
are equal to each other.

00:00:35.290 --> 00:00:36.950
Now what I want to
do in this video

00:00:36.950 --> 00:00:40.040
is show what I want to prove.

00:00:40.040 --> 00:00:43.600
So what I want to prove here
is that these two-- and they're

00:00:43.600 --> 00:00:47.330
sometimes referred to as base
angles, these angles that

00:00:47.330 --> 00:00:49.860
are between one of the
sides, and the side that

00:00:49.860 --> 00:00:52.540
isn't necessarily equal to
it, and the other side that

00:00:52.540 --> 00:00:54.487
is equal and the side
that's not equal to it.

00:00:54.487 --> 00:00:56.070
I want to show that
they're congruent.

00:00:56.070 --> 00:01:02.630
So I want to prove
that angle ABC, I

00:01:02.630 --> 00:01:05.269
want to prove that that
is congruent to angle ACB.

00:01:10.560 --> 00:01:12.130
And so for an
isosceles triangle,

00:01:12.130 --> 00:01:14.370
those two angles are
often called base angles.

00:01:14.370 --> 00:01:17.040
And this might be called
the vertex angle over here.

00:01:17.040 --> 00:01:19.470
And these are often
called the sides

00:01:19.470 --> 00:01:22.280
or the legs of the
isosceles triangle.

00:01:22.280 --> 00:01:23.995
And these are--
obviously they're sides.

00:01:23.995 --> 00:01:26.190
These are the legs of the
isosceles triangle and this one

00:01:26.190 --> 00:01:27.650
down here, that
isn't necessarily

00:01:27.650 --> 00:01:30.720
the same as the other two,
you would call the base.

00:01:30.720 --> 00:01:33.440
So let's see if
we can prove that.

00:01:33.440 --> 00:01:35.470
So there's not a
lot of information

00:01:35.470 --> 00:01:37.380
here, just that these
two sides are equal.

00:01:37.380 --> 00:01:39.010
But we have, in
our toolkit, a lot

00:01:39.010 --> 00:01:40.990
that we know about
triangle congruency.

00:01:40.990 --> 00:01:43.780
So maybe we can construct
two triangles here

00:01:43.780 --> 00:01:44.580
that are congruent.

00:01:44.580 --> 00:01:46.710
And then we can use
that information

00:01:46.710 --> 00:01:49.990
to figure out whether this
angle is congruent to that angle

00:01:49.990 --> 00:01:50.697
there.

00:01:50.697 --> 00:01:52.780
And the first step, if
we're going to use triangle

00:01:52.780 --> 00:01:56.010
congruency, is to actually
construct two triangles.

00:01:56.010 --> 00:01:58.190
So one way to construct
two triangles is let's

00:01:58.190 --> 00:02:00.670
set up another point
right over here.

00:02:00.670 --> 00:02:03.350
Let's set up another
point D. And let's just

00:02:03.350 --> 00:02:07.560
say that D is the midpoint of
B and C. So it's the midpoint.

00:02:07.560 --> 00:02:10.039
So the distance
from B to D is going

00:02:10.039 --> 00:02:11.959
to be the same thing
as the distance-- let

00:02:11.959 --> 00:02:13.500
me do a double slash
here to show you

00:02:13.500 --> 00:02:15.270
it's not the same
as that distance.

00:02:15.270 --> 00:02:16.770
So the distance
from B to D is going

00:02:16.770 --> 00:02:20.110
to be the same thing as
the distance from D to C.

00:02:20.110 --> 00:02:23.560
And obviously, between any two
points, you have a midpoint.

00:02:23.560 --> 00:02:29.010
And so let me draw segment AD.

00:02:29.010 --> 00:02:31.130
And what's useful about
that is that we have now

00:02:31.130 --> 00:02:34.610
constructed two triangles.

00:02:34.610 --> 00:02:40.170
And what's even cooler is that
triangle ABD and triangle ACD,

00:02:40.170 --> 00:02:41.582
they have this
side is congruent,

00:02:41.582 --> 00:02:43.290
this side is congruent,
and they actually

00:02:43.290 --> 00:02:45.320
share this side right over here.

00:02:48.030 --> 00:02:57.390
So we know that
triangle ABD we know

00:02:57.390 --> 00:02:59.440
that it is congruent
to triangle ACD.

00:03:03.630 --> 00:03:07.640
And we know it because
of SSS, side-side-side.

00:03:07.640 --> 00:03:10.960
You have two triangles that have
three sides that are congruent,

00:03:10.960 --> 00:03:12.400
or they have the same length.

00:03:12.400 --> 00:03:14.990
Then the two triangles
are congruent.

00:03:14.990 --> 00:03:17.830
And what's useful about that
is if these two triangles are

00:03:17.830 --> 00:03:20.150
congruent, then their
corresponding angles

00:03:20.150 --> 00:03:21.380
are congruent.

00:03:21.380 --> 00:03:24.000
And so we've actually
now proved our result.

00:03:24.000 --> 00:03:30.370
Because the corresponding
angle to ABC in this triangle

00:03:30.370 --> 00:03:34.700
is angle ACD in this
triangle right over here.

00:03:34.700 --> 00:03:46.260
So that we then know that angle
ABC is congruent to angle ACB.

00:03:46.260 --> 00:03:47.790
So that's a pretty neat result.

00:03:47.790 --> 00:03:49.570
If you have an
isosceles triangle,

00:03:49.570 --> 00:03:52.920
a triangle where two of
the sides are congruent,

00:03:52.920 --> 00:03:55.360
then their base angles,
these base angles,

00:03:55.360 --> 00:03:57.715
are also going to be congruent.

00:03:57.715 --> 00:03:59.340
Now let's think about
it the other way.

00:03:59.340 --> 00:04:00.673
Can we make the other statement?

00:04:00.673 --> 00:04:02.070
If the base angles
are congruent,

00:04:02.070 --> 00:04:05.120
do we know that these two legs
are going to be congruent?

00:04:05.120 --> 00:04:06.940
So let's try to
construct a triangle

00:04:06.940 --> 00:04:09.180
and see if we can
prove it the other way.

00:04:09.180 --> 00:04:12.050
So I'll do another
triangle right over here.

00:04:12.050 --> 00:04:15.370
Let me draw another
one just like that.

00:04:15.370 --> 00:04:17.410
That's not that
pretty of a triangle,

00:04:17.410 --> 00:04:19.700
so let me draw it
a little nicer.

00:04:19.700 --> 00:04:22.630
I'm going to draw it like this.

00:04:22.630 --> 00:04:24.260
Let me do that in
a different color.

00:04:24.260 --> 00:04:27.820
So I'll call that A.
I will call this B. I

00:04:27.820 --> 00:04:30.750
will call that C
right over there.

00:04:30.750 --> 00:04:32.970
And now we're going to
start off with the idea

00:04:32.970 --> 00:04:39.260
that this angle, angle ABC,
is congruent to angle ACB.

00:04:39.260 --> 00:04:41.854
So they have the
same exact measure.

00:04:41.854 --> 00:04:43.520
And what we want to
do in this case-- we

00:04:43.520 --> 00:04:45.470
want to prove-- so let
me draw a little line

00:04:45.470 --> 00:04:48.670
here to show that we're
doing a different idea.

00:04:48.670 --> 00:04:51.524
Here we're saying if these
two sides are the same,

00:04:51.524 --> 00:04:53.440
then the base angles are
going to be the same.

00:04:53.440 --> 00:04:54.240
We've proved that.

00:04:54.240 --> 00:04:55.080
Now let's go the other way.

00:04:55.080 --> 00:04:56.460
If the base angles
are the same, do we

00:04:56.460 --> 00:04:58.010
know that the two
sides are the same?

00:04:58.010 --> 00:05:10.240
So we want to prove that
segment AC is congruent to AB.

00:05:10.240 --> 00:05:13.094
Or you could say that
the length of segment AC,

00:05:13.094 --> 00:05:14.510
which we would
denote that way, is

00:05:14.510 --> 00:05:18.470
equal to the length
of segment AB.

00:05:18.470 --> 00:05:21.029
These are essentially
equivalent statements.

00:05:21.029 --> 00:05:21.570
So let's see.

00:05:21.570 --> 00:05:25.567
Once again in our toolkit, we
have our congruency theorems.

00:05:25.567 --> 00:05:27.150
But in order to apply
them, you really

00:05:27.150 --> 00:05:29.200
do need to have two triangles.

00:05:29.200 --> 00:05:31.470
So let's construct
two triangles here.

00:05:31.470 --> 00:05:33.950
And this time, instead
of defining another point

00:05:33.950 --> 00:05:41.935
as the midpoint, I'm going to
define D this time as the point

00:05:41.935 --> 00:05:43.980
that if I were to
go straight up,

00:05:43.980 --> 00:05:46.700
the point that is
essentially-- if you view

00:05:46.700 --> 00:05:51.300
BC as straight horizontal, the
point that goes straight down

00:05:51.300 --> 00:05:54.380
from A. And the reason
why I say that is there's

00:05:54.380 --> 00:05:59.290
some point-- you could
call it an altitude-- that

00:05:59.290 --> 00:06:02.000
intersects BC at a right angle.

00:06:02.000 --> 00:06:05.470
And there will definitely
be some point like that.

00:06:05.470 --> 00:06:07.220
And so if it's a right
angle on that side,

00:06:07.220 --> 00:06:08.940
if that's 90
degrees, then we know

00:06:08.940 --> 00:06:11.880
that this is 90 degrees as well.

00:06:11.880 --> 00:06:14.520
Now, what's
interesting about this?

00:06:14.520 --> 00:06:15.690
And let me write this down.

00:06:15.690 --> 00:06:31.870
So I've constructed AD such
that AD is perpendicular to BC.

00:06:34.470 --> 00:06:36.250
And you can always
construct an altitude.

00:06:36.250 --> 00:06:39.990
Essentially, you just have to
make BC lie flat on the ground.

00:06:39.990 --> 00:06:41.990
And then you just have
to drop something from A,

00:06:41.990 --> 00:06:43.980
and that will give you
point D. You can always

00:06:43.980 --> 00:06:46.410
do that with a
triangle like this.

00:06:46.410 --> 00:06:49.320
So what does this give us?

00:06:49.320 --> 00:06:52.220
So over here, we have
an angle, an angle,

00:06:52.220 --> 00:06:54.420
and then a side in common.

00:06:54.420 --> 00:06:57.350
And over here, you have an angle
that corresponds to that angle,

00:06:57.350 --> 00:06:59.016
an angle that corresponds
to this angle,

00:06:59.016 --> 00:07:01.760
and the same side in common.

00:07:01.760 --> 00:07:05.000
And so we know that these
triangles are congruent by AAS,

00:07:05.000 --> 00:07:07.280
angle-angle-side,
which we've shown

00:07:07.280 --> 00:07:10.190
is a valid congruent postulate.

00:07:10.190 --> 00:07:19.300
So we can say now
that triangle ABD

00:07:19.300 --> 00:07:21.810
is congruent to triangle ACD.

00:07:26.430 --> 00:07:29.860
And we know that by
angle-angle-side.

00:07:29.860 --> 00:07:31.610
This angle and this
angle and this side.

00:07:31.610 --> 00:07:33.756
This angle and this
angle and this side.

00:07:33.756 --> 00:07:35.880
And once we know these two
triangles are congruent,

00:07:35.880 --> 00:07:40.090
we know that every corresponding
angle or side of the two

00:07:40.090 --> 00:07:42.220
triangles are also
going to be congruent.

00:07:42.220 --> 00:07:45.680
So then we know that AB is
a corresponding side to AC.

00:07:45.680 --> 00:07:49.930
So these two sides
must be congruent.

00:07:49.930 --> 00:07:59.370
And so you get AB is going
to be congruent to AC,

00:07:59.370 --> 00:08:02.110
and that's because these
are congruent triangles.

00:08:02.110 --> 00:08:03.800
And we've proven what
we wanted to show.

00:08:03.800 --> 00:08:05.934
If the base angles are
equal, then the two legs

00:08:05.934 --> 00:08:06.850
are going to be equal.

00:08:06.850 --> 00:08:09.266
If the two legs are equal,
then the base angles are equal.

00:08:09.266 --> 00:08:12.050
It's a very, very, very
useful tool in geometry.

00:08:12.050 --> 00:08:14.730
And in case you're curious,
for this specific isosceles

00:08:14.730 --> 00:08:18.720
triangle, over here we set
up D so it was the midpoint.

00:08:18.720 --> 00:08:21.420
Over here we set up D so
it was directly below A.

00:08:21.420 --> 00:08:23.290
We didn't say whether
it was the midpoint.

00:08:23.290 --> 00:08:25.873
But here, we can actually show
that it is the midpoint just as

00:08:25.873 --> 00:08:28.130
a little bit of a bonus
result, because we

00:08:28.130 --> 00:08:30.710
know that since these two
triangles are congruent,

00:08:30.710 --> 00:08:34.870
BD is going to be
congruent to DC

00:08:34.870 --> 00:08:36.780
because they are the
corresponding sides.

00:08:36.780 --> 00:08:38.480
So it actually
turns out that point

00:08:38.480 --> 00:08:43.400
D for an isosceles triangle,
not only is it the midpoint

00:08:43.400 --> 00:08:47.780
but it is the place where,
it is the point at which AD--

00:08:47.780 --> 00:08:53.640
or we could say that AD is a
perpendicular bisector of BC.

00:08:53.640 --> 00:08:58.930
So not only is AD perpendicular
to BC, but it bisects it.

00:08:58.930 --> 00:09:02.860
That D is the midpoint
of that entire base.