section 72 is on similar polygons to polygons are similar if and only if their corresponding angles are congruent and their corresponding sides are proportional that means that they have the same shape but are not the same size so the angles are going to be the same but the sides are not going to be the same length they're just going to be proportional and we'll take see what that looks like so here we have quadrilateral ABCD and quadrilateral e F G H and we use the single little squiggly it's kind of like what we put on top of the congruent sign to mean is similar to so quadrilateral ABCD is similar to quadrilateral EFG H now the order that we state those letters does matter actually it doesn't matter for the first one I can name my first quadrilateral however I want so I can say ABCD which kind of makes sense since that's an alphabetical order but if they're gonna correspond to each other then a and E needs to be in the same spot be an F need to be in the same spot C and G and D and H so quadrilateral ABCD is similar to quadrilateral EFG H if and only if angle a is congruent to angle E maybe I'll go ahead and Mark those angle a is congruent to angle E angle B is congruent to angle F angle C is congruent to angle G and angle D this is getting a little sloppy here is congruent to angle H notice they correspond to each other a is the first letter mentioned here E is the first letter mentioned here so they are congruent to each other and we need to say the sides are proportional there's no way to mark this on the triangle so we just have to write proportion here without the ratios so a B over EF whatever number that ends up being they're not equal to each other because a B in this case is longer than EF a B divided by EF is the same as B see / FG which is the same as CD / G H which is the same as ad / eh so this is what we mean by proportional notice the angles are congruent and the sides are proportional corresponding sides are proportional so let's take a look at an example here if triangle ABC is similar to triangle d EF then lists all of the pairs of congruent angles and write a proportion that relates the corresponding sides so I know the triangle ABC is similar to triangle d EF they're not congruent they're not the same size but they do have the same shape now just looking at these two triangles doesn't look like they're the same shape because it looks like actually to get d EF kind of in the same orientation I think we'd have to flip it and turn it but we can also work off of this similarity statement right here since a is the first letter here and D is the first letter here I can say that angle a is congruent to angle D and you can actually mark that in the triangle if you think that's going to help continuing to work off of that B is the middle one he is the middle one so angle B is congruent to angle E and C and F are the last ones angle C is congruent to angle F and let's just go ahead and mark those B is congruent to E and C is congruent to F okay so that's the list of the pairs of congruent angles now I need to write a proportion that relates the corresponding sides might be a little hard to look at it in the diagram here since the one triangle is kind of flipped around but I can actually even just work off of the letters in the similarity statement so a B which are the first two letters of the first one over D F I'm sorry de so first two over first two is equal to BC so that's the second and the third over EF which is the second the third and then I need a third one so I'm going to go with the first letter in the last letter AC and again the first letter and last letter D F so this is showing that these two triangles are similar to each other because all of their angles are congruent and their corresponding sides are proportional and this would be the proportion of showing the corresponding sides scale factor is the ratio of the lengths of corresponding sides so in the previous example let's go back up here if the length of a B were 6 and the length of de were 3 and because a b and de are corresponding to each other okay then we could say that the scale factor of ABCD because remember a B is 6 and de is 3 so ABCD 2 d EF is going to be a B over de 6 over 3 reducing that 2 over 1 or usually with scale factor we write it like this with a colon a ratio of 2 to 1 so what this is basically saying is that ABCD is twice as big as de F now notice the order matters if I said the scale factor of ABC d a.b a.b c - d EF i'm going to do 6 - 3 and the scale factor of d EF remember de was 3 and a B was 6 in this case it's going to be de over a be sick 3 over 6 1 over 2 or 1 to 2 so the way they ask you for the scale factor matters which one - which one let's do an example here determine whether the pair of figures is similar if so write the similarity statement in other words say this triangle is similar to that triangle and find the scale factor if not explain why they're not okay let's take a look at the picture here so angles are the easiest way to start so angle a is 117 angle H is 117 okay so angle a is congruent to angle H angle B is 27 angle F is 27 so angle B is congruent to angle F and angle C is 36 and angle J is 36 so angle C is congruent to angle J so corresponding angles are congruent now our corresponding sides congruent let's check that so I know what AC is an AC since this is the first in the last would correspond to HJ so is AC over let's the AC over H J equal to let's do a B and a B would correspond to H F and is that equal to BC and BC would correspond to F J in other words are all of these fractions equal to each other let's see so AC / HJ would be 6 over 3 here a B / H F would be 8 over 4 and BC is 12 and FJ is 6 so are all of those fractions equal to each other indeed they are because they all come out to be 2 okay so yes these triangles are similar so we have angles congruent and the sides are proportional therefore they are similar triangles so let's write a similarity statement remember it doesn't matter how I name the first triangle but I do have to be pay attention then to the way I named the second triangle so I'm gonna for the first one I'm just gonna go with triangle ABC is similar to triangle now if I did ABC I have to do H F J because angle a is congruent to angle H B is congruent to F and C is congruent to J so that's my similarity statement when it says write a similarity statement that's what I'm looking for now find a scale factor now notice in this problem it didn't say the scale factor of this triangle to that triangle so I generally if they don't state it I generally go from the one on the left to the one on the right so the scale factor of triangle ABC - triangle h FJ is going to be basically any one of these here AC over H J all of these proportions are the same so that's going to be two to one so each of these fractions reduces reduces to two over one so those are the two parts of my answer here we have pentagon ABCDE and Pentagon are s t u v and we need to find x and y okay we know that they're similar to each other so I know that a B corresponds to RS I know that II D corresponds to V U and C D corresponds to tu so I'll go ahead and write that out so a B over RS is equal to e d over V U which is equal to C D over T you notice one other thing when I wrote my proportion all of my letters on the top refer to the first Pentagon and all of my letters on the bottom refer to the second Pentagon now to make this a little simpler I'm going to add in some numbers so a B is 6 RS is for Edie is 8 and vu is y plus 1 CD is X and T U is 3 so basically we have to find x and y and this is one kind of long proportion but I can actually break it up into two different ones so I can look at these two right here okay because that's going to give me a proportion I can do that has only Y in it and there's a little sloppier here I can make this one here can I do this there we go how about that okay and then those two are gonna be equal to each other I'll go ahead and write that out so it looks a little better so six over four is going to equal x over three and six over four is going to equal eight or Y plus one now we have some pretty straightforward proportions to solve so for x equals 18 or X is equal to 9 over 2 or you can write it as 4 point 5 I'll accept either one of those and then doing the cross bar up here 6 times y plus 1 is 32 6 y plus 6 equals 32 6 y plus 26 or Y is equal to 13 over 3 and I'm going to leave that like that since that's going to give me a repeating decimal okay so just set up your proportion with corresponding sides to find the x and the y and here we have pentagon ABCDE and Pentagon r s t u v we need to find the scale factor and the perimeter of each polygon okay so I'm going to go ahead and fill in since I know that AE and Edie are congruent I'm going to make that afford I don't I do know that a b and c d are congruent but i don't know what that is and bc is just 6 v r and v u are congruent so that's a 7 UT and RS are congruent so that's a 10.5 what I don't know and maybe I'll go ahead and use some letters here is I don't know what a B and C D are and I don't know what st is so I can give those letters make sure they're different letters since they are different lengths okay so I can go ahead and set up some proportions here it's kind of nice that they line up and then one of them isn't twisted so I can say 4 over 7 which is AE over r v is equal to a B which is x over 10.5 and then I need to find what Y is and that corresponds to BC which is six over Y okay so let's go ahead and solve each one of those so I'll do this first one so 7x is equal to 42 so X is equal to 6 so let me go ahead and fill in what X was so X was 8 B and C D that's equal to 6 and then doing my other proportion so I'm going to use this for over 7 and 6 over Y that gives me 4 y equals 42 or Y is equal to 10.5 well that's kind of crazy so those sides are congruent to each other so Y corresponding to s T so sto is 10.5 okay so to find the perimeter of each polygon so the perimeter Pentagon ABCDE and what did we figure out six is there X X is equal to 6 so this is a six and this is a 6b and let me go ahead and do the other one - this was a 10.5 so it is going to be 4 plus 4 plus 6 plus 6 plus 6 or 26 that perimeter Pentagon rst UV is gonna be seven plus seven was ten point five actually just could have written this as ten point five times three but go ahead and add them all up and that ends up being forty five point five okay and then we have to know the scale factor so that's the perimeter of each polygon twenty six and forty five and a half and it says find the scale factor scale factor is just going to be any two sides corresponding to each other and since they didn't distinguish between which two which you know which polygon which Pentagon to which Pentagon I'm gonna go from the left to the right so the scale factor is seven to four and actually this is kind of interesting if you were to do 26 over forty five point five that is also going to give you four over 7 so the scale factor that uses the corresponding sides is also going to work when the perimeters correspond to each other