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# Intro to quadrilateral

CCSS.Math:

There are many different types of quadrilaterals. A quadrilateral is defined as a two-dimensional shape with four sides, four vertices, and four angles. There are two main types: concave and convex. There are also various subcategories of convex quadrilaterals, such as trapezoids, parallelograms, rectangles, rhombi, and squares. Created by Sal Khan.

## Video transcript

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.