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# Proof: Opposite sides of a parallelogram

CCSS.Math:

## Video transcript

well we're going to prove in this video it's a couple of fairly straightforward parallelogram related proofs and this first one we're going to say if we have this parallelogram ABCD let's prove that the opposite sides have the same length so prove that a B is equal to DC and that ad is equal to BC so let me draw a diagonal here so I'm going to draw a diagonal 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 could do a better job so nope that's not any better that is about as good as I can do so if we look if we view D be this diagonal D be we can view it as a transversal for the parallel lines a B and D C 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 it's going to be congruent to angle B DC because they are alternate into your angles you have a transversal parallel lines so we know that angle a BD is going to be congruent to angle B DC angle B DC now you can also view this diagonal D be you could view as a transversal of these two parallel lines of the other to the other pair of parallel lines ad and BC and if you look at it that way then you immediately see that angle DBC angle DBC right over here angle D BC is going to be congruent to angle ADB to angle ADB and we'll a DB 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 alternate interior angles anterior 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 CD eebee both share this side over here it's obviously equal to itself now what is 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 a DB is congruent to triangle nan labeled to pink to green C C BD c BD and this comes out of angle-side-angle congruency so this is from angle side angle angle side angle congruence 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 side DC corresponds to side B a side DC on this bottom triangle corresponds to side B a on that top triangle so they need to be congruent so DC so we get DC is going to be equal to VA and that's because they are corresponding sides corresponding sides of congruent congruent triangles so this is going to be equal to that and by that exact same logic ad ad corresponds to C B ad corresponds to C B ad is equal to C B 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 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 let me that's the hardest part let's see draw it that's pretty good alright so we obviously know that C B is going to be equal to itself so I'll draw it like that we all 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 they have both 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 triangle a and we're starting a and then I'm going to the one hash side so a C a CB is congruent to triangle DBC D BC and this is by by side side side side side side congruence well what does that do for us what tells us that all of the corresponding angles are going to be congruent so for example ABC angle ABC is going to be so let me mark that angle a b c is going to be congruent you can say ABC is going to be congruent to d CB d CB angle d CB and you could say by you could say 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 d CB so these two angles are going to be congruent well this is interesting because here you have a line and it's intersecting a B and C D and we clearly see that these these things that could be alternate angles alternate interior angles are congruent and because we have these congruent alternate interior angles we know that a B must be parallel to C D so this must be parallel to that so we know that a B is parallel to CD by alternate 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 angle a CB is congruent to angle DBC to angle DBC angle D BC and we know that by corresponding corresponding angles congruent of congruent triangles so we're just saying that 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 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 are opposite sides have the same length and if opposite sides have the same length and 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 can say if opposite sides are parallel of a quadrilateral or you could say opposite sides of 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 only if their lengths are equal are they parallel we've proven it in both directions