Quadrilateral overview Basics of quadrilaterals including concave, convex ones. Parallelograms, rectangles, rhombi and squares
- What I wanna do in this video is give an overview of quadrilaterals.
- And you can imagine from this prefix, or I guess you could say from the beginning of this word - quad
- This involves four of something.
- And quadrilaterals, as you can imagine, are, are shapes.
- And we're gonna 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.
- Here you can measure. Here actually let me draw this one a little bit bigger 'cause 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, 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, it has an interior angle that is larger than 180 degrees.
- So, for example, this interior angle right over here is larger, is larger than 180 degrees.
- 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, "Well, 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 are 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,
- this 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 gonna focus on convex quadrilaterals
- so that's gonna be all of this space over here.
- So one type of convex quadrilateral is a trapezoid.
- 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
- 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 A, B, C, D,
- we could say that segment AB is parallel to segment DC
- and because of that we know that this is, 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 say, 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 a 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 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 as 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. If I have a quadrilateral, and if I have two pairs of parallel sides
- So two of 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.
- They 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.
- So if the four sides, so from parallelograms, these are, this is all in the parallelogram universe.
- What I'm drawing right over here, that is all the parallelogram universe.
- This parallelogram tells me that opposite sides are parallel.
- And 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 a rectangle,
- or of any, of any quadril, of any quadrilateral, is actually a hund- 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
- a 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 we do have, where we do have the length of all the sides be 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 lengths.
- 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 this is the universe of rectangles
- So the universe of rectangles. Drawing a little 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 would definitely be a parallelogram.
- It would be a parallelogram. Four right angles.
- Four right angles, and all the sides would have the same length.
- And you probably. 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 are both rhombi, are are members of the, they can also be considered a rhombus
- and they can 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.
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