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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.

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  • boggle blue style avatar for user Sabriel Holcom
    What's the difference between Angle Angle Side and Angle Side Angle?
    (16 votes)
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  • starky sapling style avatar for user LoganS
    If the velocity of a raccoon equals the width and height of a chair how does that soccer payer score a goal?
    (18 votes)
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    • duskpin tree style avatar for user WRaven
      well, If we know that the chair's angular hypotenuse is divisible 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.
      (14 votes)
  • hopper happy style avatar for user Rahul6511
    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?
    (16 votes)
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    • stelly orange style avatar for user USER_R_04
      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. 🙂👍
      (6 votes)
  • blobby green style avatar for user Ella
    why couldn't you use the two congruent sides of AB and DC to use the postulate SAS?
    (11 votes)
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  • cacteye blue style avatar for user braxman0910
    three seconds of silence go crazy
    (13 votes)
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  • hopper cool style avatar for user Kavin Puri
    Can someone please tell me an overview of this?? I am kinda confused. Thanks!
    (9 votes)
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    • sneak peak green style avatar for user bethany chiu
      Basically, you need to figure out what sides and angles two triangles share and determine if they are postulates for congruent triangles. For example, if two triangles have the same length for two sides and the same measure for one angle, then they are congruent because AAS and ASA are both postulates for congruent triangles. However, if you only know that the two triangles have the same angle measure for all three angles, then you would need more information to determine whether or not they are congruent.
      (1 vote)
  • starky sapling style avatar for user lilbona
    At around , 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?
    (5 votes)
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    • mr pink green style avatar for user David Severin
      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.
      (8 votes)
  • female robot grace style avatar for user 🅗🅐🅝🅝🅐🅗 😜
    The postulate and congruence theorems are the worst part of math. O_O
    I have watched so many videos and I still don't get it!
    And I absolutely hate word problems, they are my weakness! X(
    (7 votes)
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  • blobby green style avatar for user Attila
    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?
    (6 votes)
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  • aqualine ultimate style avatar for user anim-somuahda
    My question is why couldnt we prove the first triangle is congruent. CA=CA because of reflexive so that is side then the parallel lines so wont that make it ASA? Or must both triangles have all 3 measurments as in they all have 1 angle 1 side and one angle?
    (6 votes)
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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.