Main content

## High school geometry

### Course: High school geometry > Unit 3

Lesson 3: Congruent triangles- Triangle congruence postulates/criteria
- Determining congruent triangles
- Calculating angle measures to verify congruence
- Determine congruent triangles
- Corresponding parts of congruent triangles are congruent
- Proving triangle congruence
- Prove triangle congruence
- Triangle congruence review

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Proving triangle congruence

Given a figure composed of 2 triangles, prove that the triangles are congruent or determine that there's not enough information to tell. Created by Sal Khan.

## Want to join the conversation?

- What's the difference between Angle Angle Side and Angle Side Angle?(8 votes)
- Angle Angle Side is when one of the angles is opposite the side.

Angle Side Angle is when both angles are adjacent to the side.(26 votes)

- Why can't you use the alternate interior angles on the other side to prove congruency? For a better explanation, why isn't angle ACB = DAC?(12 votes)
- Why not? We can always use both alternate interior OR exterior, it's an excellent way, but you should know the variables/measurements. What I mean is that you should have the same variable in both triangles, whether it was a variable or a numerical value. I variables, you can count on 2 x Theta, for 2 congruent angles, in that case, u can prove using these variables. 🙂👍(4 votes)

- why couldn't you use the two congruent sides of AB and DC to use the postulate SAS?(11 votes)
- Because they do not say that segment AB is congruent to segment DC, they only say that they are parallel. As Sal said in the video, you can't say things are congruent just because they look like they are.(5 votes)

- If the velocity of a raccoon equals the width and height of a chair how does that soccer payer score a goal?(10 votes)
- well, If we know that the chair's angular hypotinuse is divisable by the leg length of the player's chair, we can use the AAAAAASSSASSASSSSSS congruency postulate to conclude that the racoon is doomed to eat trash for the rest of his life.

simple as that.(6 votes)

- Can someone please tell me an overview of this?? I am kinda confused. Thanks!(7 votes)
- I have no clue what the hell is going on...(6 votes)
- At around1:14, Sal says that CB and DA are not parallel, or that we couldn't assume as such. But by using the logic that AC is congruent to itself and that DC and AB are congruent, then couldn't we say that CB and DA must also be congruent?

No matter what, in the end there is only one way to connect both sides of the triangle using a straight line. So that would result in both triangles having the same side length, and they must also have the same angle measures. And if they have the same angle measures,*in the same spots*(because we know 2 side lengths + an angle and where they are), then CB and DA must also be parallel. (We don't need another angle for this, I don't think, but we could easily prove that angle DAC is congruent to ACB, too.)

Right?

Riiight?

I'm not being dumb, am I?(3 votes)- Merely because two sides of a triangle are congruent does not automatically mean the third side is congruent, it can be in a range of numbers. If one side is 4 and a second is 2, the third side could range fron 4-2<x<4+2. If the two line segments are not parallel, then the third sides would not be congruent.(6 votes)

- Isn't segment AC a transversal of sides DC and AB? If angles DCA and BAC are congruent, then why aren't angles DAC and ACB as well? Aren't they alternate interior angles of the transversal AC? Can someone please help?(5 votes)
- three seconds of silence go crazy(5 votes)
- What's the point in proving triangle congruence?(4 votes)
- If you can prove that two triangles are congruent, you know that all of their corresponding angles and sides are also congruent. Proving that the triangles are congruent would unlock this information, allowing you to apply it to discover even more about the triangles.(3 votes)

## Video transcript

- [Instructor] What I would
like to do in this video is to see if we can prove that triangle DCA is congruent to triangle BAC. Pause this video and see if you can figure
that out on your own. All right, now let's work
through this together. So let's see what we can figure out. We see that segment DC is
parallel to segment AB, that's what these little arrows tell us, and so you can view this segment AC as something of a transversal
across those parallel lines, and we know that alternate interior angles would be congruent, so we know for example that the measure of this angle is the same as the measure of this angle, or that those angles are congruent. We also know that both of these triangles, both triangle DCA and triangle
BAC, they share this side, which by reflexivity is going
to be congruent to itself, so in both triangles, we have an angle and a
side that are congruent, but can we figure out anything else? Well you might be tempted to make a similar argument thinking that this is parallel to that
'cause it looks parallel, but you can't make that
assumption just based on how it looks. If you did know that, then you would be able to
make some other assumptions about some other angles here
and maybe prove congruency. But it turns out, given the
information that we have, we can't just assume that
because something looks parallel, that, or because something
looks congruent that they are, and so based on just
the information given, we actually can't prove congruency. Now let me ask you a
slightly different question. Let's say that we did give you a
little bit more information. Let's say we told you that the measure of this angle right
over here is 31 degrees, and the measure of this angle
right over here is 31 degrees. Can you now prove that
triangle DCA is congruent to triangle BAC? So let's see what we can deduce now. Well we know that AC is in both triangles, so it's going to be congruent to itself, and let me write that down. We know that segment AC is congruent to segment AC, it sits in both triangles,
and this is by reflexivity, which is a fancy way of
saying that something is going to be congruent to itself. Now we also see that AB is
parallel to DC just like before, and AC can be viewed as
part of a transversal, so we can deduce that angle CAB, lemme write this down, I should
be doing different color, we can deduce that angle CAB, CAB, is congruent to angle ACD, angle ACD, because they are alternate,
alternate interior, interior, angles, where a transversal
intersects two parallel lines. So, just to be clear, this angle, which is CAB, is congruent to this angle, which is ACD. And so now, we have two angles and a side, two angles and a side, that are congruent, so we can now deduce by
angle-angle-side postulate that the triangles are indeed congruent. So we now know that triangle
DCA is indeed congruent to triangle BAC because of angle-angle-side congruency, which we've talked about
in previous videos, and just to be clear, sometimes people like
the two-column proofs, I can make this look a little bit more like a two column-proof by
saying these are my statements, statement, and this is my
rationale right over here. And we're done.