Topic D: Drawing, analysis, and classification of two-dimensional shapes
What I want to do in this video is give an overview of quadrilaterals. And you can imagine, from this prefix, or, I guess you could say, the beginning of this word, quad-- this involves four of something. And quadrilaterals, as you can imagine, are shapes. And we're going to be talking about two-dimensional shapes that have four sides and four vertices 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 four sides, four vertices, and, clearly, four angles. One angle, two angles, three angles, and four angles. Actually, let me draw this one a little bit bigger, because it's interesting. So in this one right over here, you have one angle, two angles, three angles, and then you have this really big angle right over there. If you look at the interior angles of this quadrilateral. Now, quadrilaterals, as you can imagine, can be subdivided into other groups based on the properties of the quadrilaterals. And the main subdivision of quadrilaterals is between concave and convex quadrilaterals. So you have concave, and you have convex. And the way I remember concave quadrilaterals, or really concave polygons of any number of shapes, is that it looks like something has caved in. So, for example, this is a concave quadrilateral. It looks like this side has been caved in. And one way to define concave quadrilaterals-- so let me draw it a little bit bigger, so this right over here is a concave quadrilateral-- is that it has an interior angle that is larger than 180 degrees. So for example, this interior angle right over here is larger than 180 degrees. And it's an interesting proof. Maybe I'll do a video. It's actually a pretty simple proof to show that, if you have a concave quadrilateral, if at least one of the interior angles has a measure larger than 180 degrees, that none of the sides can be parallel to each other. The other type of quadrilateral, you can imagine, is when all of the interior angles are less than 180 degrees. And you might say, wait-- what happens at 180 degrees? Well, if this angle was 180 degrees, then these wouldn't be two different sides, it would just be one side. And that would look like a triangle. But if all of the interior angles are less than 180 degrees, then you're dealing with a convex quadrilateral. So this convex quadrilateral would involve that one and that one over there. So this right over here is what a convex quadrilateral could look like-- four points, four sides, four angles. Now, within convex quadrilaterals, there are some other interesting categorizations. So now we're just going to focus on convex quadrilaterals, so that's going to be all of this space over here. So one type of convex quadrilateral is a trapezoid. And a trapezoid is a convex quadrilateral, and sometimes the definition here is a little bit-- different people will use different definitions. So some people will say a trapezoid is a quadrilateral that has exactly two sides that are parallel to each other. So, for example, they would say that this right over here is a trapezoid, where this side is parallel to that side. If I give it some letters here, if I call this trapezoid ABCD, we could say that segment AB is parallel to segment DC, and because of that we know that this is a trapezoid. Now I said that the definition is a little fuzzy, because some people say you can have exactly one pair of parallel sides, but some people say at least one pair of parallel sides. So if you use the original definition-- and that's the kind of thing that most people are referring to when they say a trapezoid, exactly one pair of parallel sides-- It might be something like this. But if you use the broader definition of at least one pair of parallel sides, then maybe this could also be considered a trapezoid so you have one pair of parallel sides like that and then you have another pair of parallel sides like that. So this is a question mark where it comes to a trapezoid. A trapezoid is definitely this thing here, where you have exactly one pair of parallel sides. Depending on people's definition, this may or may not be a trapezoid. If you say it's exactly one pair of parallel sides, this is not a trapezoid, because it has two pairs. If you say at least one pair of parallel sides, then this is a trapezoid. So I'll put that in a little question mark there. But there is a name for this, regardless of your definition of what a trapezoid is. If you have a quadrilateral with two pairs of parallel sides, you are then dealing with a parallelogram. So the one thing that you definitely can call this is a parallelogram. And I'll just draw it a little bit bigger. So it's a quadrilateral, and if I have a quadrilateral, and if I have two pairs of parallel sides. So the opposite sides are parallel. So that side is parallel to that side, and then this side is parallel to that side there-- you're dealing with a parallelogram. And then parallelograms can be subdivided even further. If the four angles in a parallelogram are all right angles, you're dealing with a rectangle. So let me draw one like that. This is all in the parallelogram universe, what I'm drawing right over here. This is all the parallelogram universe. So it's a parallelogram, which tells me that opposite sides are parallel. And then if we know that all four angles are 90 degrees. And we've proven in previous videos how to figure out the sum of the interior angles of any polygon. And using that same method you could say that the sum of the interior angles of any quadrilateral is actually 360 degrees. And you see that in this special case as well. But maybe we'll prove it in a separate video. But this right over here we would call a rectangle. Parallelogram-- opposite sides parallel and we have four right angles. Now, if we have a parallelogram where we don't necessarily have four right angles, but where we do have the length of the sides being equal, then we're dealing with a rhombus. So let me draw it like that. So it's a parallelogram. This is a parallelogram, so that side is parallel to that side, this side is parallel to that side. And we also know that all four sides have equal length. So this side's length is equal to that side's length, which is equal to that side's length, which is equal to that side's length. Then we are dealing with a rhombus. So one way to view it-- all rhombi are parallelograms. All rectangles are parallelograms. All parallelograms you cannot assume to be rectangles. All parallelograms you cannot assume to be rhombi. Now, something can be both a rectangle and a rhombus. So let's say that this is the universe of rectangles. So the universe of rectangles-- I'll draw a little bit of a Venn diagram here-- is that set of shapes and the universe of rhombi is this set of shapes right over here. So what would it look like? Well, you would have four right angles and they would all have the same length. So it would look like this. So it'd definitely be a parallelogram. Four right angles and all the sides would have the same length. And this is probably the first of the shapes that you learned, or one of the first shapes. This is clearly a square. So all squares could also be considered a rhombus, and they could also be considered a rectangle, and they could also be considered a parallelogram. But clearly, not all rectangles are squares, and not all rhombi are squares. And definitely not all parallelograms are squares. This one, clearly, right over here, is neither a rectangle nor a rhombi, nor a square. So that's an overview. Just gives you a little bit of taxonomy of quadrilaterals. And then in the next few videos, we can start to explore them and find their interesting properties or just do interesting problems involving them.