Angle bisectors
Angle Bisector Theorem Proof What the angle bisector theorem is and its proof
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- what i want to do first is just
- show you what the angle bisector theorem is
- and then we'll actually
- prove it for ourselves
- so i just have an arbitrary triangle right over here
- triangle A, B, C
- and what im going to do is im 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 im just gonna bisect this angle, angle ABC
- so lets 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
- tells us that the ratio between the size that aren't this bisector
- so when i put this angle bisector here it created two smaller triangles
- out of that large one
- the angle bisector theorem tells us that the ratios between the other
- sides of these two triangles that we now created
- are going to be the same
- so it tells us that the ratio of AB
- to AD is going to be equal to the ratio of BC to - you could say
- to CD,so the ratio of - i'll color code it - the ratio of -
- that, which is this, to this, to 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 kind of - once you see the ratio of that to that
- is going to be the same as the ratio of that to that
- so that's kind of a cool result
- but you cant just accept it on -on- faith
- because its a cool result!
- you want to prove it to ourselves
- and so you could imagine right over here
- we have some ratio set up
- so we're gonna prove it using similar triangles
- and unfortunate for us
- these two triangles right here -
- aren't necessarily similar!
- we cant really
- we know that these two angles
- are congruent to each other
- but we dont know whether this angle
- is equal to that angle
- or that
- we dont know we cant make any statements like that
- so in order to actually set up this type of a statement
- we'll have to construct
- maybe another triangle!
- that will be similar to one of these
- right over here
- and 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 dont worry if its not obvious to you
- is that what happens is if we can
- continue this bisector - this angle bisector
- right over here
- so lets just continue it
- it just keeps going
- on and on and on
- and lets also maybe we can construct a similar
- triangle to this triangle over here
- if we draw a line that's parallel
- to AB down here
- so lets try to do that
- So i'm just gonna say
- well you know - you could always find
- if C is not on AB you can always find a point
- that goes through or a line that goes through C
- that is parallel
- to AB
- so lets just -by definition- lets just
- create another line right over here
- and lets say lets call this point
- right over here F
- and lets just pick this line in such a way
- that FC is parallel to AB
- so this is parallel to that
- - right over there so FC is parallel to AB
- and we can 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 would be similar to each other
- so lets see that - lets see what happens
- so before we even think about similarity
- lets think about 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 can imagine that AB keeps going like that
- FC keeps going like that
- and line BD right here is a transversal
- then whatever this angle is
- this angle is going to be as well
- for alternate interior angles
- which we've talked a lot about
- when we first talked about angles with transverals
- 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 and 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
- an interesting result
- because here we have
- a situation where if we look at this larger triangle
- BFC - 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 that was kinda cool
- we just used 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 triangle and 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 showed that BC and FC
- are the same thing
- so this is going to be the same thing
- as what we wanna prove it
- and if we can prove that FC - the ratio
- of AB to AD is the same thing
- as the ratio of FC to CD were gonna be there
- because BC - we just showed - is equal to
- FC
- but lets not start with the theorem
- let's actually get to the theorem
- so FC is parallel to AB
- able to set up these - this one - isosceles triangle
- show these sides are congruent
- now lets look at some of the other angles
- here - and then we'll make ourselves
- feel good about it
- but we have this -
- - if we look at triangle ABD
- so this triangle right over here
- and 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 be
- which is a vertical angle
- with this one
- over here
- so they're congruent
- and we know if two
- tringles have two angles
- that are the same
- - actually the third ones going
- to be the same one as well
- or you could say by the
- angle - angle similarity postulate
- these two triangles are similar
- so let me right that down
- you wanna make sure you get the corresponding sides right
- we now know by angle - angle
- and im gonna start at the green angle
- that triangle B - and then the blue angle
- BDA is similar to triangle
- so once again lets start with the green angle -
- - F then go to the blue angle FDC
- and here we want to eventually get to the
- angle bisector theorem
- so we wanna look at the ratio between
- AB and AD - similar triangles - either you could 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 AB
- and this by the way - was by
- angle - angle similarity
- wanna right that down
- so now that we know
- that they're similar
- we know that the ratio of AB to AD
- is going to be equal to
- and we could even look here
- for the corresponding sides
- the ratio of AB
- the corresponding side is
- gonna be CF
- its going to equal CF
- over AD - AD is the same thing
- as CD - over CD
- and so we know that the ratio of
- AB over AD is equal to CF over CD
- but we just proved 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
- and then we're done!
- we just proven AB over AD
- is equal to BC over CD
- so there's kinda two things
- we have to do here
- - 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
- - sorry - I have a - something stuck in my throat -
- so we're able to
- using - constructing this triangle here
- we're able to both show its a similar
- and to construct this larger
- isosceles triangle to show
- look if we can find the ratio between
- two sides of this triangle
- and this one
- then that's going to be the ratio
- of this - if we can find
- the ratio of this side to this side
- its the same as the ratio
- of this side to this side
- that's analogues to showing
- that the ratio of this side to this side
- is the same
- as BC to CD
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