If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Determining similar triangles

CCSS.Math:

## Video transcript

what I want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar using some of the postulates that we've set up so over here I have triangle BDC it's inside of triangle AEC they both share this angle right over there so that gives us one angle we need to to get to angle angle which gives us similarity and we know that these two lines are parallel and we know if two lines are parallel and we have a transversal that corresponding angles are going to be congruent so that angle is going to correspond to that angle right over there and we're done we have one angle in triangle AEC that is congruent to another angle in BDC and then we have this angle that's obviously congruent to two itself that's in both triangles so both triangles have a pair of corresponding angles that are congruent so they must be similar so we can write triangle a AC e AC e is going to be similar to triangle we want to get the letters on the right order so where the blue angle is here the blue angle there is vertex B then we go to the wide angle C and then we go to the unlabeled angle right over there B C D BCD so we did that first one now let's do this one right over here this is kind of similar but at least it looks just superficially looking at it that Y Z is definitely not parallel to s T so we won't be able to do this corresponding angle argument or especially because they didn't even label it as parallel and so you want to you don't want to look at things just a bit by the way they looked you definitely want to say what am i given and what am I not given if these weren't labeled parallel we wouldn't be able to make the statement even if they looked parallel one thing we do have is that we have this angle right here that's common to the inner triangle and to the outer triangle and they've given us a bunch of sides so maybe we can use SAS for similarity meaning if we can show the ratio of the sides on either side of this angle if they have the same ratio for from the smaller triangle to the larger triangle then we can we can show similarity so let's go and we have to go on either side of this angle right over here let's look at the shorter side on on either side of this angle so the shorter side is two and let's look at the shorter side on the other side of the angle for the larger triangle well then the shorter side is on the right-hand side and that's going to have that's going to be X T so what we want to compare is the ratio between right this way we want to see is XY over X T over X T is that equal to the ratio of the longer side of we're looking relative to this angle the longer of the two not necessary the longest of the triangle though it looks like that as well is that equal to the ratio of X Z X Z over the longer of the two sides when you're looking at this angle right here on either side of that angle for the larger triangle over X s over X s and it's a little confusing because we've kind of flipped which side but I'm just thinking about the shorter side on either side of this angle in between and then the longer side on either side of this angle so these are the shorter sides for the smaller triangle and the larger triangle these are the longer sides for the smaller triangle on the larger triangle and we see X Y this is 2 X T is 3 plus 1 is 4 X Z is 3 X Z is 3 and X s is 6 so you have 2 over 4 which is 1/2 which is the same thing as 3 6 so the ratio between the shorter sides on either side of the angle and the longer sides on either side of the angle for both triangles the the ratio is the same so by SAS we know so by SAS we know that the two triangles are congruent but we have to be careful on how we state the triangles we want to make sure we get the corresponding sides so we could say that triangle and I'm running out of space here let me write it right above here we can write the triangle X Y Z X Y Z is similar is similar to triangle so we started up at X which is the vertex of the angle and we went to the shorter side first so now we want to start at X and go to the shorter side on the large triangle so X so you go to X T s x ts X Y Z is similar to X T s now let's look at this right over here so in our larger triangle we have a right angle here but we really know nothing we really know nothing about what's going on with any of these smaller triangles in terms of their actual angles even though this looks like a right angle we cannot assume it and it shares if we look at this smaller triangle right over here it shares one side with a larger triangle but that's not enough to do anything and then this triangle over here also shares another side but that also doesn't do anything so we really can't make any statement here about any kind of similarity so there's no similarity going on here if they if they gave us if they gave us well there are some there are some shared angles this guy they both share that angle the larger triangle small triangle so there could be a statement of similarity we could make if we knew that this definitely was a right angle then we could make some interesting statements about similarity but right now we can't really do can't do anything as is let's try this one out or this pair right over here so these are the first ones we've actually separated out the triangles so they've given us the three sides of both triangles so let's just figure out if the ratios between corresponding sides are a constant so let's start with the short side so the short side here is three the shortest side here is nine square roots of three so we have want to see whether the ratio of three to nine square roots of three is equal to the next longest side over here is 3 square roots of three is equal to three square roots of three over the next longest side over here which is 27 which is 27 and then see if that's going to be equal to if that's going to be equal to the ratio of the longest side so the longest side here is a six six and then the longest side over here is an 18 square roots of three 18 square roots of three so this is going to give us let's see this is three this is it let me do this in a neutral color so three we could this becomes 1 over 3 square roots of 3 this becomes 1 over square root of 3 root 3 over 9 which seems like a different number but we want to be careful here and then this right over here this becomes this is a if you divide the numerator denominator by 6 this becomes a 1 and this becomes 3 square roots of 3 so you get 1 square roots of 3 1 three root three needs to be equal to one needs to be equal to square root of three over nine which needs to be equal to one one over three square roots of 3 and at first they don't look equal but we can actually rationalize this denominator right over here we can show that one over three square roots of 3 if you multiply it by square root of three over square root of three this actually gives you in the numerator square root of three over square root of three times square root of 3 is 3 times 3 is 9 so these actually are all the same this is actually saying this is 1 over 3 root 3 which is the same thing as square root of 3 over 9 which is this right over here which is the same thing as 1 over 3 root 3 so actually these are similar triangles so we can actually say it and I'll make sure I get the order right so start with E which is between the blue and the magenta side so that's between the blue and the magenta side that is H right over here so triangle E I'll do it like this is triangle E and then I'll go along the blue side F then I go all along the blue side over here oh sorry let me do it this way actually let me just write it this way e triangle efg we know is similar to triangle so E is between the blue and the magenta side blue and magenta side that is H and then we go along the blue side to F go along the blue side to I and then you went along the orange side to G and then you go along the orange side to j so triangle EF j EF G is similar to triangle H I J by side-side-side similarity they're not congruent sides they all have just the same ratio or the same scaling factor now let's do this last one right over here so we have let's see we have an angle that's congruent to another angle right over there and we have two sides and so it might be tempting to use side angle side because we have side angle side here and even the ratios look kind of tempting because 4 times 2 is 8 5 times 2 is 10 but it's tricky here because they aren't the same corresponding sides in order use side angle side the two sides that have the same corresponding ratios that to be on either side of the angle so in this case they are on either side of the angle in this case the four is on one side of the angle but the five is not the five is not so because if this five was over here if it was over here then we could make it an argument for similarity but with this five not being on the other side of the angle it's not sandwiching the angle with the four we can't use side angle side and frankly there's nothing that we can do over here so we can't make some strong statement about similarity for this last one