Video transcript
What I want to doin this video is give an overviewof quadrilaterals. And you can imagine,from this prefix, or, I guess you could say,the beginning of this word, quad-- this involvesfour of something. And quadrilaterals, as youcan imagine, are shapes. And we're going to be talkingabout two-dimensional shapes that have four sides and fourvertices and four angles. So, for example--one, two, three, four. That is a quadrilateral,although that last side didn't look too straight. One, two, three, four. That is a quadrilateral. One, two, three, four. These are all quadrilaterals. They all have foursides, four vertices, and, clearly, four angles. One angle, two angles, threeangles, and four angles. Actually, let me draw thisone a little bit bigger, because it's interesting. So in this oneright over here, you have one angle, two angles,three angles, and then you have this really bigangle right over there. If you look at the interiorangles of this quadrilateral. Now, quadrilaterals,as you can imagine, can be subdividedinto other groups based on the propertiesof the quadrilaterals. And the main subdivisionof quadrilaterals is between concave andconvex quadrilaterals. So you have concave,and you have convex. And the way I rememberconcave quadrilaterals, or really concave polygonsof any number of shapes, is that it looks likesomething has caved in. So, for example, this isa concave quadrilateral. It looks like thisside has been caved in. And one way to defineconcave quadrilaterals-- so let me draw it a little bitbigger, so this right over here is a concavequadrilateral-- is that it has an interior angle thatis larger than 180 degrees. So for example, this interiorangle right over here is larger than 180 degrees. And it's an interesting proof. Maybe I'll do a video. It's actually a prettysimple proof to show that, if you have a concavequadrilateral, if at least one of the interior angles hasa measure larger than 180 degrees, that none of the sidescan be parallel to each other. The other type ofquadrilateral, you can imagine, is when all of theinterior angles are less than 180 degrees. And you might say, wait--what happens at 180 degrees? Well, if this anglewas 180 degrees, then these wouldn't betwo different sides, it would just be one side. And that would looklike a triangle. But if all of theinterior angles are less than 180degrees, then you're dealing with aconvex quadrilateral. So this convexquadrilateral would involve that one andthat one over there. So this right over here iswhat a convex quadrilateral could look like-- four points,four sides, four angles. Now, within convexquadrilaterals, there are some otherinteresting categorizations. So now we're just going tofocus on convex quadrilaterals, so that's going to be allof this space over here. So one type of convexquadrilateral is a trapezoid. And a trapezoid is aconvex quadrilateral, and sometimes thedefinition here is a little bit--different people will use different definitions. So some people willsay a trapezoid is a quadrilateral thathas exactly two sides that are parallel to each other. So, for example, they wouldsay that this right over here is a trapezoid, where thisside is parallel to that side. If I give it some letters here,if I call this trapezoid ABCD, we could say that segment ABis parallel to segment DC, and because of that we knowthat this is a trapezoid. Now I said that thedefinition is a little fuzzy, because some people sayyou can have exactly one pair of parallelsides, but some people say at least one pairof parallel sides. So if you use theoriginal definition-- and that's the kind of thingthat most people are referring to when they say atrapezoid, exactly one pair of parallel sides-- Itmight be something like this. But if you use the broaderdefinition of at least one pair of parallel sides,then maybe this could also beconsidered a trapezoid so you have one pair ofparallel sides like that and then you have another pairof parallel sides like that. So this is a question markwhere it comes to a trapezoid. A trapezoid is definitelythis thing here, where you have exactly onepair of parallel sides. Depending onpeople's definition, this may or maynot be a trapezoid. If you say it's exactlyone pair of parallel sides, this is not a trapezoid,because it has two pairs. If you say at least onepair of parallel sides, then this is a trapezoid. So I'll put that in alittle question mark there. But there is a namefor this, regardless of your definition ofwhat a trapezoid is. If you have a quadrilateral withtwo pairs of parallel sides, you are then dealingwith a parallelogram. So the one thing that youdefinitely can call this is a parallelogram. And I'll just draw ita little bit bigger. So it's a quadrilateral, andif I have a quadrilateral, and if I have two pairsof parallel sides. So the oppositesides are parallel. So that side isparallel to that side, and then this side isparallel to that side there-- you're dealingwith a parallelogram. And then parallelograms canbe subdivided even further. If the four anglesin a parallelogram are all right angles, you'redealing with a rectangle. So let me draw one like that. This is all in theparallelogram universe, what I'm drawingright over here. This is all theparallelogram universe. So it's a parallelogram,which tells me that oppositesides are parallel. And then if we know that allfour angles are 90 degrees. And we've provenin previous videos how to figure out the sum of theinterior angles of any polygon. And using that samemethod you could say that the sum of the interiorangles of any quadrilateral is actually 360 degrees. And you see that in thisspecial case as well. But maybe we'll proveit in a separate video. But this right over herewe would call a rectangle. Parallelogram--opposite sides parallel and we have four right angles. Now, if we have a parallelogramwhere we don't necessarily have four rightangles, but where we do have the length ofthe sides being equal, then we're dealingwith a rhombus. So let me draw it like that. So it's a parallelogram. This is a parallelogram, so thatside is parallel to that side, this side is parallelto that side. And we also know that allfour sides have equal length. So this side's length is equalto that side's length, which is equal to thatside's length, which is equal to that side's length. Then we are dealingwith a rhombus. So one way to view it-- allrhombi are parallelograms. All rectangles areparallelograms. All parallelograms you cannotassume to be rectangles. All parallelograms youcannot assume to be rhombi. Now, something can be botha rectangle and a rhombus. So let's say that this isthe universe of rectangles. So the universe ofrectangles-- I'll draw a little bit ofa Venn diagram here-- is that set of shapes andthe universe of rhombi is this set of shapesright over here. So what would it look like? Well, you would havefour right angles and they would allhave the same length. So it would look like this. So it'd definitelybe a parallelogram. Four right anglesand all the sides would have the same length. And this is probably the firstof the shapes that you learned, or one of the first shapes. This is clearly a square. So all squares could alsobe considered a rhombus, and they could also beconsidered a rectangle, and they could also beconsidered a parallelogram. But clearly, not allrectangles are squares, and not all rhombi are squares. And definitely not allparallelograms are squares. This one, clearly,right over here, is neither a rectangle nora rhombi, nor a square. So that's an overview. Just gives you a little bitof taxonomy of quadrilaterals. And then in thenext few videos, we can start to explorethem and find their interestingproperties or just do interesting problemsinvolving them.
