# 10-6 Similar Triangles.mov today we’re going to be covering ten dash six similar triangles our main idea today is to determine if two triangles are similar and then our second main idea or objective is to use proportions to find unknown lengths of similar triangles okay much of this comes from sixth grade so you may see some things that are familiar to you okay and looking forward you will be seeing this in geometry as well algebra standards would be standard too and also standard one because we are going to be using proportions and properties to calculate okay first thing is vocabulary similar figures now similar figures can also include similar triangles so figures that have the same shape and corresponding angles are congruent and corresponding side lengths are proportional so many of you are probably wondering what corresponding means corresponding means that they are matching okay and congruent means that they are equal you’ll also notice that there is a symbol for similar we use this symbol also known as a till date this symbol to represent similar so you’ll be seeing that frequently throughout the lesson today ok so moving on with that let’s show you a visual on this what does this look like ok so in this diagram I have a statement that says triangle ABC is similar to triangle XYZ ok that’s what this little mark says that’s what the triangles are for so what that really means is that a angle a matches to angle X and you’ll notice that they’re in the same location or both first here so a matches to X B matches to Y so I could say be I’m going to put a double mark on be not just to why ok and then see matches to Z I’m going to put a triple mark to show see just to Z now the reason why we put different markings also is to show that they are different from each other so that means B is not the same as a bee is not the same as C okay so you can see that they’re all different angles okay but they are the same as the corresponding one on the other triangles of matching one okay so let’s do a little bit of matching to get warmed up here two similar triangles okay which angles correspond to each other in other words which ones match in my diagram now you notice angle a matches with angle X so i’m going to right angle by the way this marking is what I use for angle okay it looks like a less than sign but we’re going to put a little angle mark in it so that stands for angle okay angle a is and this is another notation congruent which means they’re equal use a congruent sign angle a is congruent to angle X angle B is congruent to which other angle notice two marks two marks angle why anglesey is the same as angle z so you’ll notice these are the three pairs of matching angles or corresponding angles okay now the next thing in that definition of similar figures the corresponding side lengths have to be proportional which means that matching sides have to form a proportion so let’s take a look which sides match and you can also use the angle marks to help you with that okay so from one mark to two marks we have a to be so a to be matches to one to two marks X to Y and this is proportional which means that the ratio is equal to two to three marks b2c insane oh we’re starting with the same triangle y to z and then finally we’re going all the way around three two one mark c 2 a and C to a would match with Z to X and now we’ve formed a proportion now you’ll notice that this has three ratios okay so this is what we call a proportionality statement and what that means is it just says which ones will be proportional so I can cover up anyone at any time and actually form a proportion I could do these first two and cross multiply this all I could do the last two or even the first in the last and cover up that middle one okay so i can use choose any two of them to form a proportion okay all right so we’re going to go ahead and do our first example example number one we’re going to determine whether ABC triangle ABC is similar to triangle d e f now keep in mind that based on that definition of similar figures that means two things must be satisfied okay there’s two conditions number one are the angles are the angles congruent that’s the first condition we have to satisfy secondly second condition are the sides the corresponding sides proportional okay that’s our second condition okay so let’s check our the angles congruent well a has one mark and it matches with d which means a is congruent to deep these are called congruence marks so it shows that this is the same as this so check angle a is congruent to angle D okay we notice be has two marks he has two marks so angle B is congruent to angle e-check finally see has three marks and F has three marks so I know that angle C is congruent to angle f and you’ll notice that when i’m doing this i’m always starting with the same triangle okay so that way I don’t mix anything up always start with the same triangle each time okay and then lastly the corresponding sides are the sides proportional so let’s check one to two marks that would for 21 22 marks that’s 12 will that ratio equal to 2 3 marks which is six then e2 f2 to three marks is 18 well that ratio equal C to a which is five marks two three two one mark here 15 so let’s check will this work well for 2 12 I can reduce that when I reduce it it becomes one-third when I reduce this one it becomes one-third reduce this one’s one-third so don’t they all reduce down to the same fraction or same ratio yes which means yes these are all proportion ok so to answer this question because it does say justify your answer I need to make sure that I answer the question so yes these are similar triangles these are similar music notation triangles okay because the corresponding angles are I’m going to use a symbol again can your wit and the corresponding sides our proportion so that would be my formal answer for this question ok so moving on another example ok example number to determine whether the pair of triangles is similar very much like our example number one but you’ll notice this is a little different notice they give us actual angle measures so one thing we need to go back to is how many degrees are in a triangle do you recall that remember that a triangle has 180 degrees ok which means all three of its angle measures when you add up the angles inside it should come out to 180 degrees ok so why is that information going to help us on this question well if you take a look I’ve got this is 42 degrees and this is 42 degrees which means inside this triangle these two angles are congruent ok so they are the same well if these add up when i add these up 42 plus 42 will give me 84 degrees that’s just these two angles so how much is this angle here then angle q I need to find out how much that is in order to do that I know in a triangle I have 180 degrees so i’m going to subtract off what I just found so 180 degrees minus 84 degrees will give me 96 degrees which means angle Q is 96 degrees so now I need to check the other triangle well it says that T is 96 degrees and then they say these two angles are the same they’re congruent so what that means is I have a hundred eighty degrees total I take off 96 degrees for angle t what will that leave me that will give me 70 80 84 degrees right 80 54 degrees is left for two angles so if i take the 84 degrees and split it for two angles what is 84 degrees divided by 2 that will give me 42 which means this must be 42 degrees and this is 42 degrees so now that I filled in all my degree measures for all of my angles now I can figure out whether or not these are similar are they similar are all the matching angles equal to each other in other words is P equal to s we got 42 and forty two check check q is 90 60 is 90 set 96 check check r is 42 and you is 42 check so now I can make a statement triangle I’m gonna start in this corner p q r is similar to triangle make sure they match p match 2 s.s q match 2 t.t our match to you you so triangle pqr is similar to triangle st you how do we know that we know this is because corresponding angles are congruent or equal to measure okay okay so your turn this time I want you to go ahead and try try album number one ok try problem number one is on page 563 problem so moving on our next example example number three now that we know about the angle measures and we know that in order to have similar triangles we also need a half sides proportional so let’s take a look at this next example we need to find the missing measures for this example 3 so what they’re giving us is this triangle rst is already similar to triangle uvw what that means is the angles already matched so I’m going to go through and just match my angles are matches to you s matches to V and team matches to W notice I’m putting the different marks again like I said earlier ok now I’m going to go ahead and match the sides here’s a little note match the side lengths then after that we’re going to set up proportions and then solve it so let’s match your side lights so r 1 mark 22 marks is 15 s to t 2 marks of three marx’s is five ok and then three two one mark is X so this was all from one same triangle they all go on the top now i’m going to go ahead and match them to the bottom ones so 1 22 was the 15 where’s 122 on my second triangle it’s six to 23 was the five so two to three on this other triangle is the Y three two one was the X 3 2 1 is the five now this statement of proportionality that I have right now will help me set up my two proportions to solve so what we’re going to do is when you create two problems one to solve for x and one to solve for y so to solve for x I need to have X so I’m going to take the ratio with x x over 5 and then what I need to do is pick another ratio in here that has enough information so i can solve the one with the most information is 15 / 6 so using this i can cross multiply using my cross product property and solve so cross multiply i get six times x equals 15 times 5 okay i’m going to go ahead and use a little shortcut here instead of multiplying that out let me go ahead and divide by 6 so i get x equals and i’m going to go ahead and factor and reduce I know that 15 is 2 to 3 times 5 and I know that 6 is 2 times 3 so can I reduce any of these factors so I’m replacing this or place is so i noticed that i can cancel out the threes and then leaves nothing else to cancel so then i’m going to simplify i have 25 over to you can actually leave the answer as 25 halves or if you want to you could say you’ve got 12 and a half as your link 25 12 and half units so either one of those is fine okay improper fraction is probably most preferred so now it’s time for the Y so i’m going to pick out the y which is 5 / y equals which one has the most information 15 / 6 and i’m going to do the same so i get x 6 equals 15 times naj naj 15 15 is 5 times 3 8 times 5 3 times 2 so i’m going to go ahead and cancel I’ve got my five and my five I can cancel three threes I’m not sure sir so y must be too right so now it’s your turn to try a problem and this is our treasure isle 102 which is on page five hundred sixty 360 get back on the truck and then finally give that a try finally I lost two examples here example number four name the similar triangles and find the missing measure now this one’s a little tricky because if you take a look you see this big triangle but wait you don’t see is there’s a smaller triangle hidden inside I’m going to go ahead and just outline that so you can see it this small triangle inside so the best way to do this is to probably separate the triangles so I’m going to separate I know I have a big triangle which is a e and c and then i have a smaller triangle which is B DC now I’m going to label it I notice my small triangle as a bottom leg of three centimeters and then it has hypotenuse of four centimeters these are right triangles my larger triangle has a leg length of a centimeters and a hypotenuse of 10 centimeters so I’m separating it so it’s easier to match so the question asks two things number one named the similar triangles so which triangles are similar here we’ve got triangle a ii c is similar to which other triangle be and we have to match it e is right here has the right angle so it has to match d and c so that’s the first part of the question second part we have to find our missing measure so this is where we do our matching and solving so let’s go ahead and match that and then solve it so looking at this smaller diagram which sides match I’ve got ten centimeters matches to the four centimeters and I’ve got a centimeters not just enters into my cross multiplying so 10 times three equals four times a for a equals and reduce by 2 here so I get 15 / 2 or if you would like you can also say seven sinners and finally our last example last example what if they don’t give you a diagram at all suppose a lamp post is 20 feet high and casts a shadow 30 feet pull located right next to it casts a shadow of 12 feet how tall is this pole well draw a diagram draw your lamp post and then have a shadow and label it our lamp post is 20 feet and the shadow is 30 feet now next to it there is another pole using another color here suppose next to it there’s a smaller pole and this smaller whole cast a shadow of 12 feet but we don’t know the height of the scope so I’m going to call it peephole so what I’m going to do is create similar triangles you’ll notice these are two legs so this is a triangle and then you’ll notice the larger one this is one leg this is the other and this is my other tribe so which sides match on here 20 is the height of this pole height of this pulse or 20 p equals I started with my large lamp posts so I need to go to its shadow its shadow is 30 we can the shadow of the smaller in same thing cross multiply and solve so I get 20 x 12 equals 30 times p / my 30 quick nap here three go to be more times and that’s eight so my pool must be a quickie so now it’s your turn last prime tripod go ahead and try this is the third problem and this problem is located on the next page which is on 564 number 26 and bring that problem to us tomorrow with the other to see you in class