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CCSS.Math:

So we have this larger triangle
here and inside of that, we have these other
triangles, and we're given this information
right over here. That triangle BCD is congruent
to triangle BCA, which is congruent to triangle ECD. And given just this
information, what I want to do in
this drawing, I want to figure out what every
angle on this drawing is. What's the measure
of every angle? So let's see what
we can do here. So let's just start
with the information that they've actually given us. So we know that triangle
BCD is congruent to-- well, we know all of these
three triangles are congruent to each other. So, for example, BCD
is congruent to ECD, and so their corresponding
sides and corresponding angles will also be congruent. So just looking at the
order in which they're written B, vertex B
corresponds, in this triangle, BCD, corresponds
to vertex B in BCA, so this is the B
vertex in BCA, which corresponds to the
E vertex in ECD. So all-- everything that
I've done in magenta, all of these angles
are congruent, and then we also know
that the C angle. So in BCA-- sorry, BCD,
this angle right over here, is congruent to
the C angle in BCA. BCA, the C angle
is right over here, or C is the vertex
for that angle in BCA. And that is also the C angle, I
guess we could call it, in ECD. But in ECD, we're talking about
this angle right over here. So these three angles are
going to be congruent. And I think you could
already guess a way to come up with the values
of those three angles. But let's keep looking
at everything else that they're telling us. Finally, we have
vertex D over here. So angle-- so this is the last
one in where we listed-- so in triangle BCD, this
angle right over here corresponds to the A
vertex angle in BCA. So BCA, that's
going to correspond to this angle right over here. It's really the only one
that we haven't labeled yet. And that corresponds
to this angle, this vertex right over here,
that angle right over there. And just to make it
consistent, this C should also be
circled in yellow. And so we have all
these congruences, and now we can come up with some
interesting things about them. First of all, here, angle
BCA, angle BCD, and angle DCE, they're all congruent, and
when you add them up together, you get to 180 degrees. If you put them all adjacent,
as they all are right here, they end up with
a straight angle, if you look at
their outer sides. So you have, if
these are each x, you have three of
them added together have to be 180 degrees, which
tells us that each of these have to be 60 degrees. That's the only
way you have three of the same thing adding
up to 180 degrees. Fair enough. What else can we do? Well, we have these
two characters up here. They are both equal and
they add up to 180 degrees. They are supplementary,
the only way you can have two equal
things that add up to 180 is if they're both 90 degrees. So these two characters
are both 90 degrees. Or we could say this is a right
angle, that's a right angle. And this is congruent
to both of those, so that is also 90
degrees, and then we're left with these
magenta parts of the angle. And here, we could
just say, well, 90 plus 60 plus something
is going to add up to 180. 90 plus 60 is 150. So this has to be 30
degrees to add up to 180. And if that's 30 degrees,
then this is 30 degrees. And then this thing right
over here is 30 degrees. And then the last
thing-- we've actually done what we said we would do,
we found out all of the angles. We could also think
about these outer angles. So this-- or not
the outer angles, or these combined angles. So angle say AC-- or say, angle
ABE, so this whole angle we see is 60 degrees. This angle is 90 degrees,
and this angle here is 30. So what's interesting is
these three smaller triangles, they all have the exact
same angles, 30, 60, 90, and the exact same side lengths. We know that because
they're congruent. And what's interesting is when
you put them together this way, they construct this larger
triangle, triangle ABE, that's clearly not congruent. It's a larger triangle. It has different
measures for its lengths, but it has the same angles
30, 60, and then 90. So it's actually similar
to all of the triangles that it's made up of.