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Current time:0:00Total duration:8:35

Intro to angle bisector theorem


Video transcript

what I want to do first is just show you what the angle bisector theorem is and then it will actually prove it for ourselves so I just have an arbor area arbitrary triangle right over here triangle ABC what I'm going to do is I'm going to draw an angle bisector for this angle up here we could have done it with any of the three angles but I'll just do this one it'll make our proof a little bit easier so I'm just going to bisect this angle angle ABC so let's just say that that's the angle bisector of angle ABC and so this angle right over here is equal to this angle right over here and let me call this point down here let me call it point D the angle bisector theorem angle bisector or bisector theorem tells us that the ratio between the sides that aren't this bisector so when I put this angle bisector here it created two smaller triangles out of that larger one the angle bisector theorem tells us the ratios between the other sides that of these two triangles that we've now created are going to be the same so it tells us that the ratio of a be the ratio of a B to ad to ad is going to be equal to the ratio of BC the ratio of BC - you could say CD to CD so the ratio of I'll color code it the ratio of that which is this to this - this is going to be equal to the ratio of this which is that to this right over here to CD which is that over here so it's kind of once you see the ratio of that to that's going to be the same as the ratio of that to that so that's kind of a cool result but you can't just accept it on on faith because it's a cool result you want to prove it to ourselves and so you could imagine right over here we have some ratios set up so we're going to prove it using similar triangles and unfortunate for us these two triangles right here aren't necessarily similar we can't really we know that these two angles are congruent to each other but we don't know whether this angle is equal to that angle or that it we don't know we can't make any statements like that so in order to actually set up this type of a statement will have to construct maybe another triangle that will be similar to one of these right over here and one way to do it one way to do it would be to draw another line and this is a bit of this proof wasn't obvious to me the first time that I thought about it so don't worry if it's not obvious to you is it what happens is if we can continue this bisector this angle bisector right over here so let's just continue it it just keeps going on and on and on and let's also maybe we can construct a similar triangle to this triangle over here if we draw a line that's parallel to a B down here so let's try to do that so I'm just going to say well you know you could always find if C is not on a B you can always find a point that goes through our line that goes through C that is parallel to a B so let's just by definition let's just create another line right over here and let's say and let's call this point right over here F and let's just pick pick this line in such a way that FC is parallel to a B so this is parallel to that right over there so FC FC is parallel to a B and we could just construct it that way and now we have some interesting things and we did it that way so that we can make these two triangles be similar to each other so let's see that let's see what happens so before we think about similarity let's think about what some of the angles or what we know about some of the angles here we know that we have alternate interior angles so just think about these two parallel lines so I could imagine a be keep keeps going like that FC keeps going like that and line BD right here's a transversal then whatever this angle is whatever this angle is this angle is going to be as well from alternate interior angles which we've talked a lot about when we first talked about angles with transversals and all of that so these two angles are going to be the same but this angle and this angle are also going to be the same because this angle and that angle are the same this is a bisector because this is a bisector we know that angle abd is the same as angle DBC so whatever this angle is that angle is and so is this angle and that gives us kind of some kind of an interesting result because here we have a situation we have a situation where if you look at this larger triangle B FC we have two base angles that are the same which means this must be an isosceles triangle so BC must be the same as FC so BC must be the same as FC so that was kind of cool we just use the transversal and the alternate interior angles to show that these are isosceles and that BC and FC are the same thing and that could be useful because we know that we want to we have a feeling that this this triangle in this triangle are going to be similar we haven't proven it yet but how will that help us get something about BC up here but we just show that BC and FC are the same thing so this is going to be the same thing if we want to prove it if we can prove that FC the ratio of a B to ad is the same thing as the ratio of FC to CD we're going to be there because BC we just showed is equal to is equal to FC but let's not let's not start with the theorem let's actually get to the theorem so FC is parallel to a be able to set up these isosceles try this i this one isosceles triangle show these sides are congruent now so let's look at some of the other angles here and make it make our selves feel good about it well we have this if we look at triangle abd so this triangle right over here in triangle FDC we already established that they have one set of angles that are the same and then they also both abd has this angle right over here which is going to which is a vertical angle with this one over here so they're congruent and we know if two triangles have two angles that are the same actually the third one's going to be the same as well or you could say by the angle-angle similarity postulate these two triangles are similar so let me write that down you want to make sure you get the corresponding sides right we now know by angle angle and I'm gonna start at the green angle that triangle B and then the blue angle B D a is similar to triangle so once again let's start let's start with the green angle F then you go to the blue angle FDC F D C and here we want to eventually get to the angle bisector theorem so we want to look at the ratio between a B and a D a B and ad similar triangles either you can find the ratio between corresponding sides are going to be the same on similar triangles or you could find the ratio between two sides of a similar triangle and compare them to the ratio of the same two corresponding sides on the other similar triangle and they should be the same so by similar triangles we know that the ratio of a B and this is this by the way was by angle-angle similarity I want to write that down so now that we know they're similar we know that the ratio of a B to ad we know the ratio of a B to ad is going to be equal to and we could even look here for the corresponding sides the reach of a be the corresponding side is going to be CF is going to equal CF over ad ad is the same thing as C D over C D and so we know the ratio of a B to ad is equal to CF over CD CF over CD but we just prove to ourselves because this is an isosceles triangle that CF is the same thing as BC right over here so CF is the same thing as BC then we're done we've just proven a B over ad is equal to BC over CD so there's kind of two things we have to do here is one construct this other triangle that allowed us assuming this was parallel that gave us two things that gave us another angle to show that they're similar and also allowed us to establish so I have a some something stuck in my throat just coughed off-camera so I should go get a drink of water after this so we're able to using constructing this triangle here we're able to both show it's similar and to construct this larger isosceles triangle to show like if we could find a ratio between two sides of this triangle on this one then that's going to be the ratio of the this if we can fire a show of this side to this side is the same as the ratio of this side to this side that's analogous to showing that the ratio of this side to this side is the same as BC - CD and we are done