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Triangle similarity postulates/criteria

CCSS.Math:

Video transcript

let's say we have triangle ABC it looks something like this a b c i want to think about the minimum amount of information i want to come up with a couple of postulates that we can use to determine whether another triangle is similar to triangle ABC so we already know that if if all three angles all three of the corresponding angles are congruent to the corresponding angles on ABC then we know that we're dealing with congruent triangles so for example if this is 30 degrees this angle is 90 degrees and this angle right over here 60 degrees and we have another triangle that looks like this that looks like this it's clearly a smaller triangle but its corresponding angles so this is 30 degrees this is 90 degrees and this is 60 degrees we know that X Y Z in this case is going to be similar to ABC so we would know we would know from this because corresponding angles are congruent we would know that triangle ABC is similar to triangle X Y Z and you got to get the order right to make sure that you have the right corresponding angles Y corresponds to the 90 degree angle X corresponds to the 30 degree angle a corresponds to the 30 degree angle so a and X are the first two things B and Y which are the 90 degrees or the second two and then Z is the last one so that's what we know already if you have three angles but do you need three angles if we only knew two aying of the angles would that be enough well sure because if you know two angles for a triangle you know the third so for example if I have another triangle if I have a triangle that looks like this let me look at draw it like this and if I told you that only two of the corresponding angles are congruent so maybe maybe this angle right here is congruent to this angle and that angle right there is congruent to that angle is that enough to say that these two triangles are similar well sure because in a triangle if you know two of the angles then you know what the last angle has to be if you know that this is 30 and you know that that is 90 then you know that this angle has to be 60 degrees whatever you whatever these two angles are backed them from 180 and that's going to be this angle so in in general in order to show similarity you don't have to show three angles or three corresponding angles are congruent you really just have to show two so this is will be our the first of our similar similarity postulates we've called it angle angle if you could show that two corresponding angles are congruent then we're dealing with similar triangles so for example just to put some numbers here if you showed if this was 30 degrees and we know that on this triangle this is 90 degrees right over here we know that this triangle right over here is similar to that one there and you can really just go to the third angle pretty straight in it's pretty straightforward way you say this third angle is 60 degrees so all three angles are the same that's one of our constraints for similarity now the other thing we know about similarity is that the ratio between all of the sides are going to be the same so for example if we have another triangle right over here let me draw another triangle I'll call this triangle I'll call this triangle x y and z and let's say that we know that the ratio between a B and XY we know that a B over X Y so the ratio between this side and this side notice we're not saying that they're congruent we're just saying that their ratio we're looking at the ratio now we're saying a B over X Y let's say that that is equal to BC BC over Y Z that is equal to BC over Y Z and that is equal to AC that is equal to AC over X Z that is equal to AC over X Z so once again this is one of this is one of the ways that we say hey this means similarity so if you have all three corresponding sides the ratio between all three corresponding sides are the same then we know we are dealing with similar triangles so this is what we call side-side-side similarity and you don't want to get these confused with side side side congruence so these are all of our Simha clarity postulates similarity postulates or axioms or things that we're going to sue and then we're going to build off of them to solve problems and prove other things side-side-side when we talk about congruence means that the corresponding sides are congruent side side side for similarity we're saying that the ratio between corresponding sides are going to be the same so for example if this right over here if this right over here is let's say this right over here is 10 let me have no one think of a bigger number let's say this is 60 this right over here is 30 and this right over here is 30 square roots of three and I just made those numbers right because you we will soon learn what typical ratios are of the sides of 30-60-90 triangles and let's say this one over here is 6 3 & 3 square roots of 3 notice a B over X Y a B over X Y 30 square roots of 3 over 3 square roots of 3 this will be 10 this will be 10 what is B C over X Y 30 divided by 3 is 10 and what is 60 divided by 6 well or say C over X Z a C over X C well that's going to be 10 so in general to go from the corresponding side here to the corresponding side there we always multiply by 10 on every side so we're not saying they're congruent or we're not saying the sides are the same for this side side side for similarity we're saying we're really just scaling them up by the same amount or another way to think about it the ratio between corresponding sides are the same now what about what about if we had if we had let's start another triangle right over here let me draw it like this actually I want to leave this here so we can have our list so let me draw another triangle ABC let's draw another triangle ABC so this is a B and C and let's say that we know let's say that we know that this side took it when we go to another triangle when we go to another triangle that we know that XY that we know that XY is a B multiplied by some constant so eight so I can write it over here XY is equal to some constant times a B actually let me make XY bigger so it actually it doesn't have to be that constant could be less than one in which case it would be a smaller value let me just do it that way so let me just make XY look a little bit bigger so let's say that this is X and that is y so let's say that we know that XY X Y over a B over a B is equal to some constant or if you multiplied both sides by a B you would get XY is some scaled up version of a B so you know maybe this is maybe a B is 5 X Y is 10 then our constant would be 2 we scaled it up by a factor of 2 and let's say we also know we also know that angle ABC is congruent to angle X Y Z I'll add another point over here so let me draw another side right over here so this is Z so let's say we also know that angle a ABC is congruent to XYZ now let's say we know that the ratio between BC and Y Z is also this constant the ratio between BC and Y Z is also equal to the same constant so in the example where this is 5 and 10 maybe this is 3 and 6 the constant we're kind of doubling the length of the side so is this triangle is triangle XYZ going to be similar well if you think about it there's only one if you say that this is some multiple if X if X Y is this is the same multiple of a B as Y Z is a multiple of BC and this the angle in between is congruent there's only one triangle we can set up over here we're only constrained to one triangle right over here and that and so we're completely constraining the length of this side and the length of this side is going to have to be that same scale as that over there and so we call that side angle side similarity side-angle-side so once again we saw sss and SAS in our congruence postulates but we're saying something very different here we're saying that in SAS if the ratio between one course and the other corresponding one course in the ratio between corresponding sides of the truth triangle are the same so a B and XY of one corresponding side and then another corresponding side so that's that second side so that's between BC and Y Z and the angle between them are congruent then we're saying it's similar for SAS for congruence we said that the sides actually had to be congruent here we're saying that the ratio between the corresponding sides just has to be the same so for example SAS just to apply it if I have let me just show some examples here so let's say I have an angle here that is 3 to 4 and let's say we have another triangle here we have another triangle here that has length that has length 9 6 and we also know that the angle in between are congruent so that that angle is equal to that angle what SAS in the similarity world tells you is that these triangles are definitely going to be similar triangles that we're actually constraining because there's actually only one triangle we can draw right over here it's a triangle where all of the sides are going to have to be scaled up by the same amount so there's only one long side right here that we could actually draw and that's going to have to be scaled up by 3 as well there's only half this is the only possible triangle if you constrain this side if you're saying look this is 3 times that side this is 3 times that side and the angle between them is congruent there's only one triangle we could draw we could make and we know there there is a similar triangle there where everything is scaled up by a factor of 3 so that one triangle we could draw has to be that one similar triangle so this is what we're talking about SAS we're not saying that this side is congruent to that side or that side is congruent to that side we're saying that they're scaled up by the same factor if we had another triangle if we had another triangle that looked like this so maybe this is 9 this is 4 and the angle between them were congruent you couldn't say that they're similar because this side is scaled up by a factor of 3 this side is only scaled up by a factor of 2 so this one right over there you could not say that it is necessary that necessarily similar and likewise if you had a trying that had length nine here and length six there but you did not know you did not know that these two angles are the same once again you're not constraining this enough and you would not know that those two triangles are necessarily similar because you don't know those that middle angle is the same now you might be saying well there was a few other postulates that we had we had we had a a s when we dealt with congruence but if you think about it we've already shown that two angles by themselves are enough to show similarity so why worry about an angle and angle and a side or the ratio between a side so why even worry about that and we also had angle side angle and congruence but once again we already know that two angles are enough so we don't need to throw in this extra side so we don't even need this right over here so these are going to be our similarity postulates and I want to remind you side side side this is different than the side side side for congruence we're talking about the ratio between corresponding sides we're not saying that they're actually congruent and here side angle side it's it's different than the side angle side for congruence it's kind of related but we're here we're talking about the ratio between the sides not the actual measures