Quadrilateral ABCD is a rhombus
What they want us to prove is that their diagonals are perpendicular,
that AC is perpendicular to BD
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
That side is parallel to that side
These 2 sides are parallel
In a rhombus, not only are the opposite sides parallel,
but also all the sides have equal length
This side is equal to this side, which is equal to that side,
which is equal to that side right over there
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
For example, let me label this point in the center, point E
We know that AE is going to be equal to EC,
I'll put 2 slashes right over there
We also know that EB is going to be equal to ED
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're gonna prove that AC is perpendicular to BD
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 2 angles right over here correspond to each other
then they have to be the same and they'll be supplementary
and they'll be 90 degrees so let's just prove it to ourselves
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 here with a new color,
ABE is congruent to triangle CBE
Once we know that,
we know that all the corresponding angles are congruent
In particular, we know that angle AEB is going to be congruent
to angle CEB because they are corresponding angles
of congruent triangles
This angle right over here is going to be equal to
that angle over there
We also that they're supplementary
Let me write it this way
They're congruent and they are supplementary
These 2 are gonna have the same measure and
they need to add up to 180 degrees
If I have 2 things that are the same thing and
add up to 180 degrees, what does that tell me?
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're supplementary
This is a right angle and then, this is a right angle
Obviously, this is a right angle
This angle down here is a vertical angle,
that's gonna be a right angle
This is a right angle this over here is gonna be a vertical angle
You see the diagonals intersect at a 90 degree angle
so we've just proved
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