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Proof: Opposite sides of a parallelogram

Sal proves that a figure is a parallelogram if and only if opposite sides are congruent. Created by Sal Khan.

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  • blobby green style avatar for user shreyajain112
    Is AAA and AAS is appropraite property for prooving that the triangles are congruent?
    (2 votes)
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  • starky sapling style avatar for user madalyn long
    I don't understand what this guy is teaching me may someone please help me understand?
    (3 votes)
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  • duskpin ultimate style avatar for user 🚀The knowledge Hunter🔭
    Isn't the second theorem just a converse of the first theorem ?
    (2 votes)
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  • mr pants teal style avatar for user Juan Rodriguez
    I still dont quite grasp the side angle thing (SSS AAA SAS ASA ASS SSA) I watched the vids explainin them a lot and still dont get it and I c thatthey keep poppin up on the other geometry vids!!! HELP?!!!?!
    (2 votes)
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    • aqualine ultimate style avatar for user Namparkourse
      they are all just ways to show whether or not triangles are congruent. the "S" stands for "side" and "A" stands for "angle". The congruence theorem of "ASS" does not work and I'll explain it in an interesting story. Back in the old days when they were driving herds of horses and donkeys(asses), whenever they got to a bridge the horses would cross just fine but the donkeys were afraid to cross. In Euclid's proposition 5 the diagram takes the shape of a bridge and mathematicians over time called it the "Pons Asinorum" or the "bridge of asses". This was part of proving a congruence theorem and it was said that if asses could learn geometry then they would get no further than prop 5. It also shows that "angle , side ,side " congruence does not work which means "ASS" can't cross the bridge.
      (2 votes)
  • spunky sam blue style avatar for user Samir Gunic
    What do you call it when a line like DB is congruent to itself? Reflexiv? Reflexiv property? Reflexive postulate? Sal mentioned it briefly in one of the videos on triangles I think but I can't find it.
    (2 votes)
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  • blobby green style avatar for user fboard9596
    for the second problem could you use a theroum
    (3 votes)
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  • piceratops ultimate style avatar for user winnie6262003
    hi sal
    (3 votes)
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  • piceratops ultimate style avatar for user Noah Thekkumpuram
    congruent means equal or same as . Right?
    (2 votes)
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  • winston baby style avatar for user invinciblepsn
    how can we say that angle abd=bdc in the first instance?
    (2 votes)
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    • duskpin ultimate style avatar for user Shyam462
      We can say that abd=bdc because the line he drew through the parallelogram is technically a transversal. Because it is a transversal, the two angles it forms are congruent, since we already know that the lines are parallel to each other. Hope this helps :)
      (2 votes)
  • blobby green style avatar for user musharrafsalimi
    Is there a video where he explains ASA, SAS and all those terms? Also any video or practice recommendations since my regents or coming up in 10 days?
    (2 votes)
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Video transcript

What we're going to prove in this video is a couple of fairly straightforward parallelogram-related proofs. And this first one, we're going to say, hey, if we have this parallelogram ABCD, let's prove that the opposite sides have the same length. So prove that AB is equal to DC and that AD is equal to BC. So let me draw a diagonal here. And this diagonal, depending on how you view it, is intersecting two sets of parallel lines. So you could also consider it to be a transversal. Actually, let me draw it a little bit neater than that. I can do a better job. Nope. That's not any better. That is about as good as I can do. So if we view DB, this diagonal DB-- we can view it as a transversal for the parallel lines AB and DC. And if you view it that way, you can pick out that angle ABD is going to be congruent-- so angle ABD. That's that angle right there-- is going to be congruent to angle BDC, because they are alternate interior angles. You have a transversal-- parallel lines. So we know that angle ABD is going to be congruent to angle BDC. Now, you could also view this diagonal, DB-- you could view it as a transversal of these two parallel lines, of the other pair of parallel lines, AD and BC. And if you look at it that way, then you immediately see that angle DBC right over here is going to be congruent to angle ADB for the exact same reason. They are alternate interior angles of a transversal intersecting these two parallel lines. So I could write this. This is alternate interior angles are congruent when you have a transversal intersecting two parallel lines. And we also see that both of these triangles, triangle ADB and triangle CDB, both share this side over here. It's obviously equal to itself. Now, why is this useful? Well, you might realize that we've just shown that both of these triangles, they have this pink angle. Then they have this side in common. And then they have the green angle. Pink angle, side in common, and then the green angle. So we've just shown by angle-side-angle that these two triangles are congruent. So let me write this down. We have shown that triangle-- I'll go from non-labeled to pink to green-- ADB is congruent to triangle-- non-labeled to pink to green-- CBD. And this comes out of angle-side-angle congruency. Well, what does that do for us? Well, if two triangles are congruent, then all of the corresponding features of the two triangles are going to be congruent. In particular, side DC on this bottom triangle corresponds to side BA on that top triangle. So they need to be congruent. So we get DC is going to be equal to BA. And that's because they are corresponding sides of congruent triangles. So this is going to be equal to that. And by that exact same logic, AD corresponds to CB. AD is equal to CB. And for the exact same reason-- corresponding sides of congruent triangles. And then we're done. We've proven that opposite sides are congruent. Now let's go the other way. Let's say that we have some type of a quadrilateral, and we know that the opposite sides are congruent. Can we prove to ourselves that this is a parallelogram? Well, it's kind of the same proof in reverse. So let's draw a diagonal here, since we know a lot about triangles. So let me draw. There we go. That's the hardest part. Draw it. That's pretty good. All right. So we obviously know that CB is going to be equal to itself. So I'll draw it like that. Obviously, because it's the same line. And then we have something interesting. We've split this quadrilateral into two triangles, triangle ACB and triangle DBC. And notice, all three sides of these two triangles are equal to each other. So we know by side-side-side that they are congruent. So we know that triangle A-- and we're starting at A, and then I'm going to the one-hash side. So ACB is congruent to triangle DBC. And this is by side-side-side congruency. Well, what does that do for us? Well, it tells us that all of the corresponding angles are going to be congruent. So for example, angle ABC is going to be-- so let me mark that. You can say ABC is going to be congruent to DCB. And you could say, by corresponding angles congruent of congruent triangles. I'm just using some shorthand here to save some time. So ABC is going to be congruent to DCB, so these two angles are going to be congruent. Well, this is interesting, because here you have a line. And it's intersecting AB and CD. And we clearly see that these things that could be alternate interior angles are congruent. And because we have these congruent alternate interior angles, we know that AB must be parallel to CD. So this must be parallel to that. So we know that AB is parallel to CD by alternate interior angles of a transversal intersecting parallel lines. Now, we can use that exact same logic. We also know that angle-- let me get this right. Angle ACB is congruent to angle DBC. And we know that by corresponding angles congruent of congruent triangles. So we're just saying this angle is equal to that angle. Well, once again, these could be alternate interior angles. They look like they could be. This is a transversal. And here's two lines here, which we're not sure whether they're parallel. But because the alternate interior angles are congruent, we know that they are parallel. So this is parallel to that. So we know that AC is parallel to BD by alternate interior angles. And we're done. So what we've done is-- it's interesting. We've shown if you have a parallelogram, opposite sides have the same length. And if opposite sides have the same length, then you have a parallelogram. And so we've actually proven it in both directions. And so we can actually make what you call an "if and only if" statement. You could say opposite sides of a quadrilateral are parallel if and only if their lengths are equal. And you say if and only if. So if they are parallel, then you could say their lengths are equal. And only if their lengths are equal are they parallel. We've proven it in both directions.