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CCSS Math: HSG.CO.C.11

We're told that quadrilateral
ABCD is a rhombus. And what they want us to
prove is that their diagonals are perpendicular, that
AC is perpendicular to BD. Now let's think about everything
we know about a rhombus. First of all, a rhombus
is a special case of a parallelogram. In a parallelogram, the
opposite sides are parallel. So that side is
parallel to that side. These two sides are parallel. And in a rhombus, not only are
the opposite sides parallel-- it's a parallelogram--
but also, all of the sides have equal length. So this side is equal
to this side, which is equal to that side,
which is equal to that side right over there. Now, there's other
interesting things we know about the diagonals
of a parallelogram, which we know all rhombi
are parallelograms. The other way around is
not necessarily true. We know that for any
parallelogram-- and a rhombus is a parallelogram-- that the
diagonals bisect each other. So for example, let me label
this point in the center. Let me label it point E. We know that AE is
going to be equal to EC. I'll put two slashes
right over there. And we also know that EB
is going to be equal to ED. So this is all of what
we know when someone just says that ABCD is a rhombus,
based on other things that we've proven to ourselves. Now we need to prove that
AC is perpendicular to BD. So an interesting way to prove
it-- and you can look at it just by eyeballing it-- is if we
can show that this triangle is congruent to this triangle and
that these two angles right over here correspond
to each other, then they have to be the same. And they'll be supplementary,
and then they'll be 90 degrees. So let's just prove
it to ourselves. So the first thing we see
is we have a side, a side, and a side; a side,
a side, and a side. So we can see that triangle--
let me write it here. Let me [? do this ?]
in a new color. We see that triangle ABE is
congruent to triangle CBE. And we know that by
side-side-side congruency. We have a side, a side,
and a side; a side, a side, and a side. And then once we
know that, we know that all the corresponding
angles are congruent. And in particular, we
know that angle AEB is going to be
congruent to angle CEB. Because they are corresponding
angles of congruent triangles. So this angle right
over here is going to be equal to that
angle over there. And we also know that
they are supplementary. Let me write it this way. They're congruent, and
they are supplementary. These two are going to
have the same measure, and they need to add
up to 180 degrees. So if I have two things
that are the same thing and they add up to 180 degrees,
what does that tell me? So that tells me that
the measure of angle AEB is equal to the measure
of angle CEB, which must be equal to 90 degrees. They're the same measure,
and they are supplementary. So this is a right angle, and
then this is a right angle. And obviously, if
this is a right angle, this angle down here
is a vertical angle. That's going to
be a right angle. If this is a right
angle, this over here is going to be a vertical angle. And you see the diagonals
intersect at a 90-degree angle. So we've just proved--
so this is interesting. A parallelogram, the
diagonals bisect each other. For a rhombus, where
all the sides are equal, we've shown that not only
do they bisect each other but they're perpendicular
bisectors of each other.