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## Theorems concerning quadrilateral properties

# Proof: The diagonals of a kite are perpendicular

CCSS.Math:

## Video transcript

What I want to do
in this video is I want to prove that segment
AC is perpendicular to segment DB based on the
information that we have in this diagram over here. That this side has the
same length as that side, this side has the same
length as that side. And I'll give you a hint,
we're going to use one or more of our congruence postulates. I'll just stick with calling
them postulates from now on. And so the ones that
we know-- so let me draw a little line here. This is kind of our tool kit. We have the side
side side postulate, if the three sides
are congruent, then the two triangles
are congruent. We have side angle side, two
sides and the angle in between are congruent, then the two
triangles are congruent. We have ASA, two angles
with a side in between. And then we have AAS, two
angles and then a side. So any of these things
we've established-- these are our postulates--
we're going to assume that they imply congruency. And I'm also going to do this
as what we call a two column proof. And you don't have to do
something as a two column proof, but this is
what you normally see in a normal
introductory geometry class. So I thought I would
expose you to it. It's a pretty basic idea is
that you make a statement, and you just have to give the
reason for your statement. Which is what we've been
doing with any proof, but we haven't always put
it in a very structured way. So I'm just going
to do it like this. I'll have two columns like that. And I'll have a
statement, and then I will give the reason
for the statement. And so the strategy that
I'm going to try to do is it looks like,
right off the bat, it seems like I can
prove the triangle CDA is congruent to triangle
CBA based on side side side. And that's a pretty
good starting point because once I can
base congruency, then I can start to
have angles be the same. And the reason why
I can do that is because this side is
the same as that side, this side the same as that side,
and they both share that side. But I don't want to just
do it verbally this time, I want to write it out properly
early in this two column proof. So we have CD, we had
the length of segment CD is equal to the length of CB. CD is equal to CB,
and that is given. So these two characters
have the same length. We also know that DA,
the length of segment DA, is the same as the
length of segment BA. So DA is equal to BA, that's
also given in the diagram. And then we also know
that CA is equal to CA, I guess we could say. So CA is equal to itself. And it's obviously
in both triangles. So this is also given, or
it's obvious from the diagram. It's a bit obvious. Both triangles share that side. So we have two triangles. Their corresponding sides
have the same length, and so we know that
they're congruent. So we know that triangle CDA
is congruent to triangle CBA. And we know that by the
side side side postulate and the statements
given up here. Actually, let me
number our statements just so we can refer back
to this 1, 2, 3, and 4. And so side side side
postulate and 1, 2, and 3-- statements 1, 2, and 3. So statements 1, 2, and 3 and
the side side postulate let us know that these two
triangles are congruent. And then if these are congruent,
then we know, for example, we know that all of their
corresponding angles are equivalent. So for example,
this angle is going to be equal to that angle. So let's make that
statement right over there. We know that angle
DCE-- so this is going to be statement 5-- we
know that angle DCE, that's this angle right over here, is
going to have the same measure, we could even say
they're congruent. I'll say the
measure of angle DCE is going to be equal to
the measure of angle BCE. And this comes straight out
of statement 4, congruency, I could put it in parentheses. Congruency of those triangles. This implies straight,
because they're both part of this
larger triangle, they are the
corresponding angles, so they're going to have
the exact same measure. Now it seems like we could do
something pretty interesting with these two smaller
triangles at the top left and the top right of this,
looks like, a kite like figure. Because we have a side,
two corresponding sides are congruent, two corresponding
angles are congruent, and they have a side in common. They have this side in
common right over here. So let's first just
establish that they have this side in
common right over here. So I'll just write statement 6. We have CE, the measure or
the length of that line, is equal to itself. Once again, this
is just obvious. It's the same. Obvious from diagram
it's the same line. Obvious from diagram. But now we can use
that information. So we don't have three sides,
we haven't proven to ourselves that this side is the
same as this side, that DE has the
same length as EB. But we do have a side, an
angle between the sides, and then another side. And so this looks pretty
interesting for our side angle side postulate. And so we can say, by the
side angle side postulate, we can say that triangle DCE
is congruent to triangle BCE. And when I write the
labels for the triangles, I'm making sure that
I'm kind of putting the corresponding point. So I started at D, then
went to C, then to E. So the corresponding
I guess angle, or the corresponding point
or vertex I could say, for this triangle
right over here, is B. So if I start with D, I
start with B. C in the middle is the corresponding vertex
for either of these triangles, so I put it in the middle. And then they both go to E.
And that's just to make sure that we are specifying
what's corresponding to what. And we know this, we know this
is true, by side angle side. And the information
we got from-- so we got this side is established
that these two sides are congruent was from statement 1. Then that these angles are
congruent is from statement 5 right over here. And then statement 6
gave us the other side. Statement 6, just like that. And if we know that these
triangles are congruent, that means that all of
their corresponding angles are congruent. So we know, for example, that
this angle right over here is going to be congruent
to that angle over there. So let's write that down. So we know, statement number
8, the measure of angle, let's call that DEC, is equal
to the measure of angle BEC. And this comes straight
from statement 7. Once again, they're congruent. Congruency. And then we also know--
we'll make statement 9. We also know that the measure
of angle DEC-- or maybe we should just write it this way. Angle DEC and angle
BEC are supplementary. They are supplementary,
and that's kind of-- you can just look
at that from inspection but I'll write a decent--
are supplementary. Supplementary, which
means they add up, their measures add
up, to 180 degrees. And we know that because
they are adjacent and outer sides
form straight angle. And then, the next step-- if we
know that these two angles are equal to each other,
and if we know that they are
complementary-- our next step means that we could
actually deduce that they must be 90 degrees. So 10, measure of
angle DEC equals measure of angle BEC,
which equals 90 degrees. And then for the reason it might
be a little bit more involved, we could put these two
statements together. So it would be
statements 8 and 9. And then statements 8
and 9 mean that DEC, so I could write this, measure
of angle DEC plus measure of angle-- actually,
let me just, since I don't want to do
too many steps all at once, let me just take it
little bit by little bit. So let me just do
it all like this. So let me say measure of angle
DEC plus measure of angle BEC is equal to 180. And this comes
straight from point 9, that they are supplementary. And then we could
say statement-- I'm taking up a lot of
space now-- statement 11, we could say
measure of angle DEC plus measure of angle DEC
is equal to 180 degrees. And we know that
from statement 9. We know that from statement
9 and statement 8. We essentially just
took statement 9 and substituted that
BEC, the measure of BEC, is the same as the
measure of DEC. And so then, if we
want statement 12, we could say
measure of angle DEC is equal to 90, which is equal
to the measure of angle BEC. And then, this comes once again
straight out of point number 11 and 8. And what you could see, I'm
kind of taking a little bit more time, going a little bit more
granular through the steps. In some of the other proofs
I would've just said, oh, obviously this
implies this or that. And then we're done. Because if these
are 90 degrees-- so let me write the last statement. So statement 13, which is
what we wanted to prove. We wanted to prove that
AC is perpendicular to DB. So AC is perpendicular
to, what was it? AC is perpendicular
to segment DB, and it comes straight
out of point 12. And we're done. We've done a two
column proof, and we have proven that this line
segment right over here is perpendicular to that line
segment right over there. We did it with SSS,
with the SSS postulate, and the side angle
side postulate.