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Current time:0:00Total duration:13:28

Triangle congruence postulates/criteria

CCSS.Math:

Video transcript

we now know that if we have two triangles and all of their corresponding sides are the same so by side-side-side so the corresponding sides all three of the corresponding sides have the same length we know that those triangles are congruent what I want to do in this video is explore if there are other properties that we can find between the triangles that can help us feel pretty good that these that those two triangles would be congruent so side side side works what about angle angle angle so let me do that over here what about angle angle angles what I'm saying is is if let's say I have a triangle like this like I have a triangle like that and I have a triangle like this and if we know if we know that this angle is congruent to that angle if this angle is congruent to that angle which means that their measures are equal or and I should say and and that angle is congruent to that angle can we say that this is that these are two congruent triangles in the first case it looks like maybe it is at least the way I drew it here but when you think about it you can have the exact same corresponding angles being having the same measure or being congruent but you can actually scale one of these triangles up and down and still have that property for example if I had this triangle right over here this triangle right over here it looks similar and I'm using that in just the everyday language size it has the same shape as these triangles right over here and it has the same it has the same angles that angle is congruent to that angle this angle down here is congruent to this angle over here and this angle over here is congruent to this angle over here so all of the angles in all three of these triangles are the same the corresponding angles have the same measure but clearly clearly this triangle right over here is not the same it is not congruent to the other two the sides are have a very different length this side is much shorter than this side right over here this side is much shorter than that side over there and this side is much shorter over here so with just angle angle angle you cannot say that the triangle has the same size and shape it does have the same shape but not the same so this does not imply this does not imply congruence so angle angle angle does not imply congruence it what it does imply and we haven't talked about this yet is that these are similar triangles so angle angle angle implies similar let me write it over here it implies similar triangles and similar you probably are used to the word in just everyday language but similar has a very specific meaning in geometry and similar things have the same shape but not necessarily the same size so anything that is congruent because it has the same size and shape is also similar but not everything that is similar is also congruent for you so for example this triangle is similar to all of all of these triangles are similar to each other but they aren't all congruent these two are congruent if their sides are the same I didn't make that assumption but if we know that their sides are the same then we can say that they're congruent but neither of these are congruent to this one right over here because this is clearly much larger has the same shape but a different size so we don't we can't have an AAA postulate or an AAA axiom to get two congruence what about side angle side so let's try this out side angle side so let's start off with one triangle right over here so let's start off with a triangle that looks like this I have my blue side I have my pink side and I have my magenta side and let's say that I have another triangle that has this this blue side it has the same side same length as that blue side so let me draw it like that it has the same length as that blue side so that length and that length are going to be the same it's it has a congruent angle right after that so this angle and the next angle for this triangle are going to be going to have the same measure or they're going to be congruent and then the next side the next side is going to have the same length as this one over here so that's going to be the same length as this over here so it's going to be the same length and we don't know and because we only know that they have that these that the court that two of the corresponding sides have the same length and the angle between them and this is important the angle between the two corresponding sides are also have the same measure we can do anything we want with this last with this last side on this one we can essentially is going to have to start right over here you could start from this point and we can pivot it to form any triangle we want but we can see the only way we can form a triangle is if we bring this side all the way over here and close this right over there and so we can see just logically that if we have if for two triangles they have one side that has the length the same the next side has a length the same and the angle in between them so this angle let me do them the same color this angle in between them this is the angle this a is this angle and that angle it's the angle in between them this first side is in blue and the second side right over here is in pink and well it's already written in pink so we can see that if two sides are the same have the same length two corresponding sides of the same length and the corresponding angle between them they have to be congruent there's no other place to put this third side so SAS and sometimes it's once again called a postulate an axiom or if it's kind of proven sometimes it's called a theorem this does imply that the two triangles are congruent so we will give ourselves this tool in our toolkit we had the sss postulate now we have the SAS postulate two sides are equal and the angle in between them for two triangles corresponding sides and angles then we can say that it is definitely a cos are congruent triangles now what about and I'm just going to try to go through all of the different combinations here what if I have something what if I have angle side angle so let me try that so what happens if I have angle side angle so let's go back to this this one right over here so actually let me just draw a redraw a new one for each of these cases so angle side angle so I'll draw a triangle here so I have this triangle that so this would be maybe the side that would be the side let me draw the whole triangle actually first so I have this triangle let me draw one side over here and then let me draw one side over there and that this angle right over here I'll call it I'll do it in orange and this angle over here I will do it in yellow so if I have another triangle that has one side being having an equal measure so I'll use it as this blue side right over here so it has one side that has equal measure and the two angles on either side of that side are either either end of that side are the same will this triangle necessarily be congruent we're just going to try to reason it out these aren't formal proofs we're really just trying to set up what a reasonable postulates or what are reasonable assumptions we can have as we kind of in our toolkit as we try to prove other things so that angle let's call it that angle right over there it's going to be they're going to have the same measure in this triangle and if this angle right over here in yellow is going to have the same measure on this triangle right over here so regardless so I'm not in any way constraining the sides over here so this side this side right over here could have any length it could have any length but it has to form this angle with it so it could have any length and it can just go as far as it wants to go it would no way have we constrained what the length of that is and actually let me mark this off too so this is the same as this so that side can be anything we haven't constrained it at all and once again this side could be anything we haven't constrained it at all but we know it has to go at this angle so it has to go at that angle well once again there's only one triangle that can be formed this way we can say all day that this length could be as long as we want or as short as we want and this one could be as long as we wanted a short as we want but the only way that they can actually touch each other and form a triangle and have these two angles is if is if they are the exact same length as these as these two sides right over here so these two this side will actually have to be the same as that side and this would have to be the same as that side once again this isn't a proof I'd call it more of a reasoning through it or an investigation really just to establish what are reasonable baselines or axioms or assumptions or postulates that we can have so for my purposes I think a si does show us that two triangles are two triangles are congruent now let's try another one let's try angle angles let's try angle-angle-side and you know in some geometry classes and you know maybe if you have to go through an exam quickly you might memorize okay side side size implies congruence and that's kind of logical side-angle-side implies congruence and so on and so forth I'm not a fan of memorizing it it might be good for time pressure it is good to sometimes even just go through this logic if you're like wait does angle angle angle work well no I can find this case that breaks down angle angle angle if these work just try to verify for yourself that they make logical sense why they would imply a congruence now let's try angle angle side let's try angle angle angle side so once again let's say let's have a triangle over here it has some side so this one's going to be a little bit more interesting so it has some side that's the side right over there and then it has two angles so let me draw the other the other sides of this triangle I'll draw one in magenta and then one in green and it has there's two angles and then the side so let's say you have this angle you have that angle right over there actually didn't have to put a double because that's the first angle that I'm so I have that angle which we refer to is that first a then we have this angle which is that second a so if I know if I know that there's another that there's another triangle that has one side having the same length so let me draw it like that it has one side having the same length it has one angle on that side that has the same measure so it has a measure like that and so this this side right over here could be of any length this side of here could be any length we aren't constraining what the length of that side is but whatever the angle is on the other side of that side is going to be the same as this green angle right over here so for example it could be like that and then you could have a green side go like that it could be like that and have the green side go like that and if we have so the only thing we're assuming we're the only thing we're assuming is that this is the same length as this and that this angle is the same measures that angle and that this measures the same measure is that angle and this magenta line can be of any length and this green line can be of any length we in no way have constrained that but how can can we form any triangle that is not congruent to this because that the bottom line is this green line is going to have to touch this one right over there and the only way is going to touch that one right over there is if it starts is if it starts right over here it starts right over here because we're constraining this angle right over here we're constraining that angle and so it looks like angle-angle-side does indeed imply congruence so that does imply congruence so let's just do one more just to kind of try out all of the different situations what if we have and I'm running out of a little bit of real estate right over here at the bottom what if we try it out side side angle so once again draw a triangle so it has one side there it has another side there and then I don't have to do those hash marks just yet so one side then another side and then another side and it what happens if we know that there's another triangle that has two of the sides the same and then the angle after it so for example it would have that side just like that and then it has another side but we're not constraining the angle we aren't constraining we aren't constraining this angle right over here but we're constraining the length of that side so let me color code it so that blue side is at first side then we have this magenta side right over there so this is going to be the same length as this right over here but let me make it a different angle to see if I can disprove it so let's say it looks like that or actually let me make it even let me make it even more interesting let me try to make it let me try to make it like let me try to make it like that let me try to make it like that so it's a very different angle but now it has to have the same angle out here it has to have that same angle out here so it has to be roughly that angle so it actually looks like we can draw a triangle that is not congruent that has the same two sides being the same length and then an angle is different for example this is pretty much that I made this angle smaller than this angle these two sides are the same this angle is the same now but what the byproduct of that is that this green side is going to be shorter on this triangle right over here so these aren't you don't necessarily have congruent triangles with side side angle so this is this is not necessarily congruent not necessarily necessarily or similar it gives us neither congruence nor similarity