Proof - Rhombus Diagonals are Perpendicular Bisectors

Proving that the diagonals of a rhombus are perpendicular

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