Main content

## High school geometry

# Geometry proof problem: congruent segments

CCSS.Math:

Sal proves that two pairs of segments are congruent using the ASA and AAS congruence criteria. Created by Sal Khan.

## Video transcript

Let's say given this diagram right over here We know that the length of segment AB is equal to the length of AC so AB 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 the angle ABF, ABF is equal to angle ACE or you could see their measures are equal or this just implies that they are congruent so they have the same measures It's equal to angle ACE 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 BF, is whether BF has the same length as CE Does BF have the same length as CE So let's try to do that So we already know a few things I could do it two column professionally let me just do it just so that in case you have to do two column proofs in your class you can see how to do it more formally So let's try out our statements Over here I'm going to write I'm going to write my reason for the statement So let me just rewrite this kind of formal two column proof so we know AB is equal to AC so this is statement 1 and this is given We know statement 2 that angle ABF is equal to angle ACE Once again that was given Now the other interesting thing on each of these ,we have an angle and we have a side, 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 ABF and triangle ACE and they both share this vertex at A and point A is a vertex for both of these So, we could say angle A we say BA, let's call it BAF, angle BAF We could say is equal to angle BAF or we could say is equal to angle CAE That makes it little bit clear that we're dealing with two different triangles right here But it really is the exact same angle It's equal to itself right there that's our third statement and we could say that it's obvious Some people would call this the reflexive property It's obvious that an angle is equal to itself And so we could say it's obvious or we could call it maybe the reflexive property that an angle is clearly reflexive obviously equal to itself even if we labeled it different way this angle is going to be the same measure And now we have something interesting going on, we have an angle, a side, and an angle So, we end up having is that triangle so by angle-side-angle we have the triangle BAF so our statement number 4 I'm running out of space right here I'll go down right here Statement right here is triangle BAF triangle BAF Let me kind of highlight it a little more blue right here BAF so that's this entire triangle right over here And half of the trick of some of this problems is seeing the right triangle So we started with this white angle we went through the side that we knew then we went to this orange angle right over here BBA I'm sorry we started at this angle then we work to this orange angle across the side E that we know is congruent to that side over there And then we went to the side the aim of the vertex is not labeled So at triangle BAF we now know is going to be congruent Congruent to triangle we start at the white angle go to the orange angle then go to the unlabeled angle It's going be congruent to angle to triangle CAF So, this is kind of a messily drawn version but you can get the idea These two triangles are going to be congruent C A sorry CAE I should say is congruent to triangle CAE White angle, orange angle, and then the unlabeled angle on that triangle right over there And this comes straight out of angle-side-angle This comes straight out of ASA and these are the two angles and so this is the side in between so it comes out of the statements 1, 2, and 3 And so they're congruent, we know that corresponding sides are going to be congruent so we know our statement 5 We should do this a little bit neater Our statement 5, we now know that BF is equal to CE BF is equal to CE And this comes straight out of statement 4, or we could say, corresponding sides, sides congruent, corresponding sides are congruent Now let's take it up another notch Let's see if we can prove whether ED is equal to EF So let's just keep going down this and see if we can prove whether ED is equal to EF I put a question mark there 'cause we haven't necessarily proven it yet So I'm gonna prove that this little short line segment EF is equal to DF Sorry not EF is equal to DF, ED 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 congruency over that but we do already have some information here We know that BAF is congruent to CAE So we also know that this side right over here Let me do it with the color I haven't used yet Let me see I have been using a lot of the colors in my pallet so it's getting a little too So we know from these two congruent triangles that side AE Side AE which is part of CAE We know that AE is going to be equal to AF That these two sides are congruent and the reason why is 'cause they're corresponding sides of congruent triangles AF is the side opposite the white angle on BAF triangle BAF And AE is the side opposite the white angle on triangle CAE which we know are congruent So we know that AE is equal to AF And once again this comes from statement 4 and we could 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 that BA or as to say we know that AB is equal to AC that was given and so we know that EB Let me write it over here and make it a little bit messy right over here Statement 7 will give us will give us some space we know that BE is going to be equal to CF Let me write that down, we know that BE is equal to CF And why do we know that 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 BE is equal to CF? Well we know that the length of BE is equal to the length of BA minus AE or I could just say AB I could that's how I call it up here so it's equal to AB minus AE is the same thing based on these last few things that we saw As saying AC minus AF cause AB is equal to AC so that's equal to AC and AE we already showed as the same thing as AF AC minus AF and 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 1, we know it from statement 5 and we know it from statement 6 Actually we didn't need we didn't need statement 5 there Let me see we just need 1 and 6 So, let's say we need this is from 1 and 6 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 minus the little part This right over here is going to be equal to this right over here So, 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 So 8, we know that angle EDB is going to be equal to angle FDC That comes straight out of 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-white angle-side So we know that these two smaller triangles are congruent So, now we know I don't want to lose my diagram We know that triangle BED, we know so statement number 9 we know that triangle BED is congruent So, BED is this one we know that BED is congruent to triangle Now we want to use the same sides white angle yellow side then orange angle 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 angle the orange labeled angle So, you want to start C unlabeled angle orange angle so triangle CFD And this comes straight from once again, orange angle white angle side So angle-angle-side orange sorry, orange angle white angle side So this come straight angle-angle-side congruency And since we've now shown that this triangle is equal to that triangle we know that their corresponding sides are equal And then this is our homestretch We now know since these two triangles are congruent We now know that ED is equal to DF because they're corresponding sides And I could write that right over here ED is equal to DF And once again the reason here is the same thing up here corresponding so we know our statement 9 which means they're congruent And 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 each triangle and you eventually get it But 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 could figure out BE by subtracting it from AE minus BE Seeing that there are two triangles kind of overlapping in the star or arms or whatever you might want to call it