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## Geometry (all content)

### Course: Geometry (all content) > Unit 11

Lesson 4: Working with triangles# Corresponding angles in congruent triangles

We write the letters of congruent triangles so their order tells us which parts are corresponding. In this video, Sal uses this notation to solve some angles. Created by Sal Khan.

## Want to join the conversation?

- At2:52how can we be sure that the three angles equal 180 degrees? It may appear to be a straight line, but without any notation ensuring that it is, how can we be sure?(13 votes)
- Hi J,

Right at the very beginning of the video, at0:01, Sal uses the words "in this larger triangle here" as he outlines triangle ABE. If this was a straight word question however, the question would start out with something like, "Given triangle ABE, prove that ..."

Hope that helps!(15 votes)

- what if it does not give that information? how do you know if they are all congruent or not?(4 votes)
- no sometimes the assignment is to figure out the congruent.(6 votes)

- Don't we need to be given a few more pieces of information? Without being told on the diagram:

Can we actually assume that A, C, and E are colinear?

Same for B, D, and E.

Do we need to assume that C is the midpoint of AE?

Same for D on BE.

And do we need to assume that CD is the altitude of triangle BCE (therefore perpendicular to BE)?

*Edit: Took the word "assume" out of first sentence.(5 votes)- youcant asuume anything in geometry(1 vote)

- こんいちは！

あのう、わかりません。

Hello!

I do not understand.

I need help with everything.

I have no clue what I'm doing, or what to do.

I struggle with math a lot and I'm not so bright when it comes to math.

Can you help me, please?(3 votes)- If we know two figures are congruent, then we know matching parts are congruent. So lets use two of the three triangles triangle BCD is congruent to triangle ECD. Letters in the same spot (first, middle, and last) give congruent angles. So <B is congruent to <E, <C is congruent to <C (note they are different angles in the two triangles, but both at vertex C) and <D congrent to <D (once again, two different angles with common vertex). Similarly, matching sides are congruent, so BC congruent of EC (first two letters), CD is congruent to CD (any length is always congrent to itself), and DB is congruent to DE. So given a congruence statement, we can match angles and sides.(4 votes)

- why does ~ on top of = mean congruent(4 votes)
- Someone created it and it stuck. The ~ alone means similar. Websites attribute it as noted below, the symbol did not transfer from the website correctly:

A similar looking symbol, º, meaning “congruent,” was credited to Johann Gauss in 1801. From http://mathserver.neu.edu/~bridger/U201/History_of_Math_Mathematical_Symbols.htm(3 votes)

- I may have forgotten from a previous video, but how do you recognize, without a protractor or ruler, what the corresponding angles are? I know they appear when there are two or more congruent triangles, but do you eye them out? Or is their a mathematical way to know. Thanks(5 votes)
- although it's obvious enough to look at it, how do we know that the large shape ABE is definitely a triangle? is there a proof that a shape made of 3 congruent triangles must also be a triangle (and not some extreme quadrilateral where angle ACE could be not 180, but 179.99 degrees or something)?(5 votes)
- Does the order of how the triangles are written matter?(2 votes)
- Very much so - corresponding angles should match on congruency statements.

For example if we say △ ABC ≅ △ EFG, that tells us a lot about the triangles, ∠ A ≅ ∠ E, ∠B ≅ ∠F, ∠C ≅ ∠G, AB ≅ EF, BC ≅ FG, CA ≅ GE.(6 votes)

- If you split a triangle into 3 smaller triangles that are congruent to each other, would they ALWAYS be similar to the large/master triangle?(3 votes)
- Not always. Here's just one example. If you split an equilateral triangle into 3 congruent triangles, they will not be equilateral (or similar) anymore.(3 votes)

- Maybe this is a silly question...but is Sal saying that the farthest left triangle is congruent to the two other? If so, could someone please explain why.

Thanks🙂(2 votes)- it only explains the information given in the question.(4 votes)

## Video transcript

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.