So, we have a parallelogram right over here
So, what we wanna prove is that it's diagonals bisect each other
So, the first thing we can think about; these aren't just diagonals,
these are lines that are intersecting parallel lines
So, you can also view them as transversals
And if we focus on DB right over here, we see that it intersects DC
and AB and it sits there
Those we know are parallelograms
We know that they are parallel
This is a parallelogram
Alternate interior angles must be congruent
So, that angle must be equal to that angle there
Let me make a label here
Let me call that middle point E
So, we know that angle ABE must be congruent to angle CDE
by alternate interior angles of
a transversal intersecting parallel line
Alternate interior angles
If we look at diagonal AC or we should call it transversal AC
we can make the same argument
It intersects here and here
These two lines are parallel
So, alternate interior angles must congruent
So, angle DEC must be --- let me write this down ---
angle DEC must be congruent to angle BAE
by for the exact same reason
Now we have something interesting
If we look this top triangle over here and this bottom triangle,
we have one set of corresponding angles that are congruent
We have a side in between that's going to be congruent
Actually, let me write that down explicitly
We know and we've proved this to ourselves in the previous video
that parallelograms not only are opposite sides are parallel they're
also congruent
So, we know from the previous video that that side is equal
to that side
So, let me go back to what I was saying
We have two sets of corresponding angles that are congruent
We have a side in between that's congruent
And then we have another set of corresponding angles
that are congruent
So, we know that this triangle is congruent to that triangle
by angle-side-angle
So, we know that triangle --- I'm gonna go from the blue
to the orange to the last one
Triangle ABE is congruent to triangle blue, orange
and the last one, CDE by angle-side-angle congruency
Now what is that do for us
What we know if two triangles are congruent, all of their
corresponding features especially all of the corresponding
sides are congruent
So, we know that side EC corresponds to EA
Or I could say side AE, we could say side AE,
corresponds to side CE
They're corresponding sides of congruent triangle
So, their measures or their lengths must be the same
So, AE must be equal to CE
Let me put two slashes since I already used one slash over here
let me focus on this -- we know that BE must be equal to DE
Once again they're corresponding sides of two congruent triangles
so they must have the same length
So, this is corresponding sides of congruent triangles
So, BE is equal to DE
And we've done our proof
We've showed that, look, diagonal DB is splitting AC into two
segments of equal length and vice versa
AC is splitting DB into two segments of equal lengths
So, they are bisecting each other
Now, let's go the other way around
Let's prove to ourselves that if we have two diagonals
of a quadrilateral that are bisecting each other that we're
dealing with a parallelogram
So, let me see
So, we're gonna assume that the two diagonals
are bisecting each other
So, we're assuming that that is equal to that
And that that, right over there, is equal to that
Given that we wanna prove that this is a parallelogram
And to do that we just have to remind ourselves
We just have to remind ourselves that this angle is going
to be equal to that angle
One of the first things we learn because they're vertical angles
So, let me write this down
C -- label this point -- angle CED is going to be equal to
or is congruent to angle, so I started is BEA, angle BEA
And that, what is that, well that shows us that these
two triangles are congruent 'cause we have a corresponding sides
of a congruent and angle in between and on the other side
So, we now know that the triangle, I'll keep this in yellow,
triangle AEB is congruent to triangle DEC by side-angle-side
congruency, by SAS congruent triangles
Fair enough
Now, if we know that two triangles congruent we know that all
corresponding sides and angles are congruent
So, for example, we know that angle CDE is going to be congruent
to angle BAE
And this is just corresponding angles of congruent triangles
And now we have this kind of transversal of these two lines that
could be parallel if the alternate interior angles are congruent
And we see that they are
These two are kind of candidate alternate interior angles and
they are congruent
So, AB must be parallel to CD
So, AB, let's just draw one arrow, AB must be parallel to CD
by alternate interior angles congruent of parallel lines
I'm just writing in some short hand, forgive the cryptic nature
of it although I'm saying it out
And so we can then do the exact same -- while we just shown
that these two sides are parallel -- we can do that exact same
logic to show that these two sides are parallel
I won't necessarily write it all out
It's exact same proof to show that these two
So, first of all, we know that this angle is congruent to that angle
right over there
And then we know, actually let me write it out, we know
that angle AEC is congruent to angle DEB, I should say
They are vertical angles
And that is the reason up here as well
Vertical angles
And then we see that triangle AEC must be congruent
to triangle DEB by side-angle-side
So, then we have triangle AEC must be congruent to triangle
DEB by SAScongruency
Now, we know that corresponding angles must be congruent
So, that we know that angle, so, for example angle CAE
must be congruent to angle BDE and this is the corresponding
angles of congruent triangles
So, CAE, let me use a new color
CAE must be congruent to BDE
And now we have a transversal
The alternate interior angles are congruent
So, the two lines that the transversals are intersecting
must be parallel
So, this must be parallel to that
So, then we have AC must be parallel to BD
by alternate interior angles
And we're done
We've just proven that if the diagonals bisect each other,
if we start that as a given then we end at a point where we say,
"Hey, the opposite sides of this quadrilateral must be parallel
or that ABCD is a parallelogram "