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Geometry proof problem: congruent segments

CCSS.Math:

Video transcript

let's say given this diagram right over here we know that the length of segment a-b is equal to the length of AC so a B which is this whole side right over here the length of this entire side as a given is equal to the length of this entire side right over here so that's the entire side right over there and then we also know that angle a bf a bf is equal to angle AC or you could see their measures are equal or this is implies that they're congruent or they have the same measures and it's equal to angle a c e so this angle right over here is congruent to that angle right over there or you could say that they have the same measure now the first thing that I want to attempt to prove in this video is whether is whether B F is whether B F has the same length as C e does bf have the same length as C e so let's try to do that so we already know a few things I could do a two column proof actually let me just do it just so that in case you have to do two column proofs in your class you can kind of see how to do it more formally so let's write our statements statement and then over here I'm going to write I am going to write my reason for the statement so let me just rewrite this just have a formal two column proof so we know a B is equal to AC so this is statement one and this is given we know statement to that angle a bf is equal to angle a C II once again that was given now the other interesting thing we have an angle and we have a side on each of these each of these triangles and then what you can see is both of the triangles and when I say both of the triangles I'm talking about triangle a B F and triangle a c e and they both share this vertex at a at point a is a vertex for both of these so we could say we could say angle angle a we say be a let's call it be AF angle be AF we could say is equal to angle be AF or we could say it's equal to angle CAE angle C AE and that makes it a little bit clearer that we're dealing with that we're dealing with two different triangles right here but it really is the exact same angle it's equal it's equal to itself right there that's our third statement and we could say it's obvious some people would call this the reflexive property that that's obvious that an angle is equal to itself and so we could say it's obvious obvious or we could call it maybe the reflexive property that an angle is clearly reflexive obviously equal to itself even if we label it different ways this angle is going to be the same measure and now we have something interesting going on we have an angle aside and an angle an angle aside and an angle so what we end up having is that triangle so by angle side angle we have the triangle be a F so our statement number four I'm running out of space right here statement number I'll go down here statement number here's triangle baf triangle baf and let me kind of highlight it in a little blue right here be a F so that's this entire triangle right over here and half of the trick of some of these problems is just seeing the right triangle so we started with this wide angle we went through the side that we knew and we went to this orange angle right over here b b:a i'm sorry we started at this angle then we went to this orange angle across the side a that we know is congruent to that side over there and then we went to the side the angle the vertex that's not labeled so be a at triangle baf we now know is going to be congruent congruent to triangle we start at the wide angle go to the orange angle then go to the unlabeled angle it's going to be congruent to angle to triangle c AF c AF so this is kind of a messily drawn version but they could get the idea these two triangles are going to be congruent see a sorry see a E I should say it's congruent to triangle see a E white angle orange angle and then the unlabeled angle in that triangle right over there and this comes straight out of this comes straight out of angle side angle this comes straight out of a si and this is one angle and this is or so this is the side in between and these are the two angles so it comes out of statements one two and three and so they are congruent we know that corresponding sides are going to congruent so we know our statement five I should do this a little bit neater our statement five we now know that B F is equal to C e VF is equal to C EB F is equal to C E and this comes straight out of statement four and we could say corresponding sides corresponding sides sides congruent sides the corresponding sides are congruent now let's take it up another notch let's see if we can prove let's see if we can prove whether II D is equal to e F so we let's just keep going down this and see if we can prove whether II D is equal to EF I put a question mark there because we haven't necessarily proven it yet so I'm gonna prove that this little short line segment II D is equal to sorry not EF is equal to DF e D is equal to DF e D is equal to DF so let's see if we can prove this right over here so the interesting thing that we might at first it might not be so obvious you know how do we figure out some type of congruence over that but we do already have some information here we know that these that BA F is congruent to CAE so we also know we also know that this side right over here let me do it in a color that I haven't used yet let me see I have not I've been using a lot of the colors and my palates are getting a little too so we know that from these two congruent triangles that side AE side AE which is part of CAE we know that we know that a e is going to be equal to AF that these two sides are congruent and the reason why is they're their corresponding sides of congruent triangles a F is the side opposite the white angle on ba f triangle baf and AE is the side opposite the white angle on triangle CAE which we know are congruent so we know that a e is equal to AE is equal to AF and once again this comes from statement 4 and we can even say corresponding sides congruent same reason as we gave right up here now what's interesting here is you know this isn't even a triangle that we're seeing up here but this information that these two characters are congruent help us with this part over here because we know we know that VA we know that or I should say we know that a b is equal to we know that a b is equal to AC that was given and so we know we know that e b let me write it over here and i'll make it a little bit a little messy right over here statement seven I'll give us will give us some space we know that B E is going to be equal to C F so let me write that down we know that V E is equal to C F and why do we know that so let me put the reason right over here let me try to clean up my work a little bit this column has been slowly drifting to the left but how do we know that V E is equal to CF well we know that b b:e the length of B E is equal to the length of B a minus AE is equal to the length of V ay minus VA minus AE minus I should say a B I could just that's how I call it up here so it's equal to a B a B minus AE which is which is the same thing based on these last few things that we saw as saying AC minus AF because a b is equal to AC so that's equal to AC and AE we already showed is the same thing as AF AC minus AF and AC AC minus AF is the same thing as CF right over here is equal to CF right over there and we know that because and we know this from statement one we know it from statement one we know it from statement five and we know it from statement and we know it from statement six actually we didn't need we didn't need statement five there let me see we just need one and six so let's say we need this is from one and six is what we had to do there so we just know that look this side is equal to that side this little part is equal to that part so if you subtract the big part - the little part this right over here is going to be equal to this right over here that's all we're showing so this yellow side is equal to this yellow side right over here now the other thing that we know and this is straight out of vertical angles is that this angle EDB is going to be congruent to angle FDC so let me write that down again eight we know that angle EDB angle e DB is going to be equal to angle F D C is going to be equal to angle F DC that comes straight out of vertical angles vertical angles are vertical angles are congruent or their measures are equal and now all of a sudden we have something interesting again we have the orange angle wide angle side orange angle white angle side so we know that these two smaller triangles are congruent so now we know and I don't want to lose my diagram we know that triangle B edie we know so statement number nine we know that triangle B II D is congruent so ve D is this one we know that ve D is congruent to triangle and we want to use the same size white angle yellow side then orange angle Y dang white angle white angle let me be careful here white angle so B is white angle E is the unlabeled angle and then D is the labeled ain't the orange label angle C well so we want to start C unlabeled angle orange Langille so C FD so triangle C FD and this comes straight from once again orange angle white angle side so angle angle side orange angle sorry orange angle white angle side so this comes straight out of angle angle side congruence and since we've now shown that this triangle is equal to that cut triangle we know that their corresponding sides are equal and then this is our homestretch we now know we now since these two triangles are congruent we now know that Edie Edie is equal to D F because their corresponding sides and I could write that right over here II D is equal to D F and once again the reason here is the same thing up here corresponding so we know we know our statement nine which means they're congruent and corresponding sides corresponding sides congruent and we are done so that was a pretty involved problem but you see once again you go step by step just try to figure out whatever you can about each triangle and you eventually get in there really the hard part isn't so much the realizing which postulate to use or how to apply them necessarily but seeing the triangle seeing that there's some information there seeing that you can figure out B by subtracting it from from by subtracting AE from a be seeing that there are two triangles kind of overlapping in this star without arms or whatever you might want to call it