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CCSS.Math:

what I want to do in this video is give an overview of quadrilaterals and you can imagine from this prefix or the I guess you could say the beginning of this word quad this involves four of something and quadrilaterals as you could imagine are our 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 that lasts I didn't look too straight one two three four that is a quadrilateral one two three four these are all quadrille arms they all have four sides four vertices and clearly four angles one angle two angles three angles and four angles here you could measure 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 angle 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 con cave and you have convex and the way I remember concave quadrilaterals or airily 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 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 what all of the interior angles are less than 180 degrees and you might say what happens at 180 degrees well if this angle was 180 degrees and these wouldn't be to 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 context cut a 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 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 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 a-b 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 the 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 parallel 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 pet 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 parallel o parallel o parallelogram parallelogram and I'll just draw it a little bit bigger so it's a quadrilateral if I have a quadrilateral and if if I have two pairs of parallel sides so two or 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 parallelogram so these are this is all in the parallelogram universe what I'm drawing right over here this is all the parallelogram universe so the parallelogram 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 it same method you could say that the sum of the interior angles of a rectangle of any quadrilateral is actually a huh it's 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 we do have where we do have the lengths of all 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 outside 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 sides length is equal to that sides length which is equal to that sides length which is equal to that size 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 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 definitely be a parallelogram it would be a parallelogram for right angles for 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 members of the they can 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 or squares or and not all rhombi are squares and definitely not all parallelograms are squares this one clearly right over here is neither a rectangle nor rhombi nor a square so that's an overview just gives you a little bit of text 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