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Determining congruent triangles

CCSS.Math:

Video transcript

but we have drawn over here is five different triangles and what I want to do in this video is figure out which of these triangles are congruent to which other of these triangles and to figure that out I'm just over here going to write our our triangle congruence postulate so we know that two triangles are congruent if all of their sides are the same so side side side we also know they are congruent if we have a side and then an angle between the sides and then another side that is congruent so side angle side if we reverse the angles on the sides we know that's also a congruence postulate so if we have an angle and then another angle and then the side in between them is is congruent then we also have two congruent triangles and then finally if we have an angle and then another angle and then aside then that is also any of these imply congruence so let's see or congruent to triangle so let's see what we can figure out right over here for these triangles so right in this triangle ABC or over here we're given this length seven then 60 degrees and then forty degrees or another way to think about it we're given an angle an angle and a side 40 degrees in 60 degrees then seven and in order for something to be congruent here they would have to have an angle angle side given at least unless we maybe we have to figure it out some other way but I'm guessing for this problem they'll just already give us the angle so they'll have to have an angle an angle side and it can't just be any angle angle inside it has to be forty sixty and seven and it has to be in the same order it can't be 60 and then 40 and then seven if the forty degree side has if one of its side has the length seven then that is not the same thing here here the 60-degree side has length seven so let's see if any of these other triangles have this kind of forty sixty degrees and then the seven right over here so this has the forty degrees in the sixty degrees but the seven is in between them so this looks like it might be congruent to some other triangle maybe closer or something like angle side angle because they have an angle side angle so it wouldn't be that one this one looks interesting this is also angle side angle so maybe these are congruent but we'll check back on that we're still focused on this one right over here and this one we have a sixty degrees then a forty degrees and a seven this is tempting we have an angle an angle and a side but the angles are in a different order here it's 40 67 here it's 60 47 so it's an angle and angle inside but the side is not on the on the 40 degree on the 60 degree angle it's on the 40 degree angle over here so this doesn't look right either here we have 40 degrees 60 degrees and then 7 so this is looking pretty good we have this side right over here is congruent to this side right over here then you have your you have your 60-degree angle right over here 60-degree angle over here it's a little it might not be obvious because it's flipped and they've they're drawn a little bit different we should never assume that just the drawing tells you what's going on and then finally you have your 40 degree angle here which is your 40 degree angle here so we can say we can write down that and I'll do it let me think of a good place to do it I'll write it right over here we can write down the triangle a b c is congruent to triangle now we have to be very careful with how we name this we have to make sure that we have the corresponding the corresponding vertices map up together so for example we started this triangle at vertex a so point a right over here that's where we have the 60 degree angle that's the vertex of the 60 degree angle so the vertex of the 60 degree angle over here is point n so I'm going to go to N and then we went from A to B B was the side was the vertex that we did not have any angle for and we could figure it out if these two guys add up to 100 then this is going to be the 80 degree angle so over here the a degree angle is going to be M the one that we don't have any label for it's kind of the other side it's the thing that shares the 7 the 7 length side right over here so then we want to go to n then M and then finish up sorry n m and M and then finish up the triangle oh no and I want to really stress this that we have to make sure we get the order of these right because then we're kind of referring to we're not showing the corresponding vertices in each triangle now we see vertex a or point a maps to point and on this congruent triangle vertex B Maps to point em and so you can say look a B the length of a B is congruent to nm so it all matches up and we can say that these two are congruent by angle angle side by a AAS so we did this one is this one right over here is congruent to this one right over there now let's look at these two characters so here we have an angle forty degrees aside in between and then another angle so it looks like a si is going to be involved we look at this one right over here we have a forty degrees 40 degrees seven and then 60 you might say wait here the forty degrees is on the bottom then here it's on the top but remember things can be congruent if you can flip them if you can flip them rotate them shift them whatever so if you flip this guy over you will get this one over here and that would not have happened if you had flipped this one to get this one over here so you see these two by you have let me just make it clear you have the 60 degree angle is congruent to this 60 degree angle you have the side of length seven is congruent to this side of length seven and then you have the 40 degree angle is congruent to this forty degree angle so once again these two characters are congruent to each other and we can write I'll write it where I have some I'll write right over here we can say triangle d EF triangle d EF is congruent to triangle and here we have to be careful again D Point D is the vertex for the 60 degree side so I'm going to start at H which is the vertex of the 60 degree side over here is congruent to triangle H and we went from D to e e is the vertex on the forty degree side kind of the other vertex that shares the seven length segment right over here so we want to go from H to G a gee I and we know that from angle-side-angle by by angle-side-angle so that gives us that that character right over there is congruent to this character right over here and then finally we're left with this poor poor chap and it looks like it is not congruent to any of them it is tempting to try to match it up to this one especially because the angles here on the bottom and you have the seven side over here angles here on the bottom and the seven side over here but it doesn't match up because the order of the angles aren't the same you don't have the same corresponding angles if you try to do this little exercise where you map everything to each other you wouldn't be able to do it right over here and this over here you know it might have been a trick question where maybe if you did the math if this was like a forty or sixty degree angle maybe then maybe you could have you could have matched this to some of the other triangles or maybe even some of them to each other but this last angle in all of these cases forty plus 60 is is a hundred this is going to be an eighty degree angle right over they have to add up to 180 this is an eighty degree angle this is an eighty degree angle if this ended up by the math being a forty or sixty degree angle then it could have been a little bit more interesting there might have been other other other congruent pairs but this is an eighty degree angle in every case the other angle is 80 degrees so this is just a loan unfortunately for for him he is not able to find a a congruent companion