Videos

8.2 & 8.3 Notes

Added Feb 16, 2016 •

Copy this URL:
Embed code:	<iframe class="edlio-embed-player" type="text/html" width="640" height="360" src="//hs.cantonisd.net/apps/embed/?v=103955" frameborder="0" allowfullscreen></iframe> Change dimensions

Show Transcript

8.2 and 8.3 we're going to do together in Italia can prove that triangles are similar to each other and I know the last section we did Polly in general and we're going to focus just on triangles not remember when polygons are similar to things have to be true and remember that the corresponding Ingles  have to be congruent and the corresponding sides have to be proportional now  all three angles in the triangles have to be congruent corresponding and all the sides have to have the same ratio or beep torsional know what if you don't know every angle and every side is there another way to tell Will if you'll recall when we did congruent singles we had five ways to prove triangles were congruent to each other without knowing every single angle inside and if you'll remember those were side what side side angle side angle side angle angle angle side and hypotenuse leg now this lesson is very similar since that but it's proving that they are similar not congruent so instead of the angles and the sides being congruent to show congruence we're going to show that the angles are congruent but the sides have the same proportion so just like proving triangles are congruent there are to prove the triangles are similar there are going to be three ways or theorems to prove triangles are similar first ways AA similarity or angle angle similarity now angles didn't work when we were doing congruence but they weren't for similarity in says if two angles of one triangle are congruent to two angles of another triangle they have to be congruent then the triangles are similar so you look here remember we had a third angle theorem that said if these two are congruent to these two the third angles had to be congruent to because they all have to add up to 180 so the minimum that you only need is two angles here congruent to two Banks cuz if we have to we know we have three so if you'll look here if I can run to D and B is congruent then these triangles have to be similar now look at this example can these triangles be proven similar by AA similarity if so write a Clarity statement No look at this we know that the right angles automatically are congruent does anything else have to be congruent look we have vertical angles so right there we have an angle in an angle in this triangle congruent to an angle and angle in this tray so it is yes and it's going to be by a similarity can you have to write the similarity so let's match up the triangles if I do ABC for the first one what matches with the a they would be the M with the B would be okay and then see matches up with the c so it would be Casey okay now let's look at this one right here now the first thing is we know that these two lines are parallel we know we have a vertical angle pair then since these are parallel what kind of angle pair is D & T that would be alternate interior so we know this angle is congruent to this angle and likewise you could have instead of D&T done in in and pee because those are alternate interior angles also a remember you only need two in each triangle you can mark all three if you want to so this one is yes now if I do triangle DPA so I go DPA what matches up with d t dust would have to go TN for the second one ten hey  the second way is SSS similarity and that stands for side-side-side soon and it says if the corresponding side lengths of two triangles are proportional  then triangles are similar so it says the corresponding side lengths that means all three sides are proportional two different triangles here in here now if you look here I have little symbols Mart this Square goes with that square side the circle goes with the circle side and the triangle goes with the triangles so if we match up all of those if we match up the ad in the RS right here you match up the AC in the RT right here and you match up the BC in the St right here those should all give you the exact same ratio so a 2 to 1 ratio 3 to 2 that means that the triangles have to be similar if the all their sides have the same ratio so let's look at number three can the two triangle keep Reason by SSS similarity if so write a similarity statement so let's leave these two triangles first of all how do I know which side is the matchup will think about it the smallest has to go with the smallest one right and then the medium seven has to go with the medium side 5 and then the biggest side eight will go with the biggest sit so make a ratio out of all of those for over 2.57 over 5 and then over 6 now remember whichever trying will you start with that has to be on top all three now what all of those equal will for over 2.5 doesn't take nicly reduce but if you put it in decimal format 1.6 7.5 in decimal form is 1.4 + 8 which we know is 1.3 or 4/3 so look these do not have the same ratio so this is no not similar because their sides don't have the same ratio the third and last way is SAS similarity set stands for side angle side where is Arabs and it says if an angle of one triangle is congruent to an angle of a second triangle and the looks of the sides including this angle are proportional then the triangles are similar so if you have one angle congruent in two sets of sides that have the same Ratio or proportion those triangles have to be congruent now the other thing I want to point out it says the sides included the two sides in the angle included between at the two sides of the angle so if you'll see here this side goes with this side and the circle goes with the Waco so you have side angle side again so remember you have to have one angle set congruent and two sides have the same race yet and that mean the triangles have to be similar to each other and in our example it says can the two triangles be proven similar by SSS SAS similarity so write a similarity statement look at these what we definitely have the angles now let's look at the site and remember the smaller matches up with the smaller the bigger with the bigger so what is 4/6 + what is 6 / 9 are those equal will that is two thirds in this equals two thirds so yes have the same race yet so we have side angle side matchup with side angle side here therefore the triangles are congruent now if I name this in let's do bat what corresponds with bat Sobe is connected to the larger side so that's in men Hume he hit  now we have some examples to do in remember we have angle angle similarity wait you can do SIM for short side angle side similarity and we have side side side similarity those are the only ones that work for similarity now you need to put that similarity or SIM for short so you distinguish it between the congruent theorems that we have now once you prove two triangles are similar you know all their angles are congruent and you know all their sides are proportional you can use that and ratio proportion to for anything you need to find so let's look at number five and six it says are these triangles similar if so State how meeting one of these three and write a similarity so let's Aaliyah number 5 do you see the two triangles their CBD and their CAE a triangle within a triangle now what they you tell us is that these lines are parallel to each other so if these are parallel we have consecutive interior but we also have this thing on single Witcher corresponding and what do we know about corresponding the Regal so that angle is congruent to that angle and likewise on this in that angle is congruent to that angle there's enough that you need for a similarity but let me point out since they both have see in them that's a reflex property seized the same angle in both triangles it's BCD and it's a c e so that one can be marked to so that would be our third angle if you needed it but this answer would be yes and it's angle angle similarity or SIM for short and triangle a he is the big one is congruent to Triangle now remember a B in the little triangles the same so it's going to be be now look at number six they don't give us angles they give us sides this time so let's look for that SSS similarity no match up your side here 10 is the smallest over here 15 is the smallest then 14 is the medium do you want is the medium and a teen's going to go with the 27th so if I do 10 / 1514 / 21 and 18 / 27 do those all equal the same amount well 10 or 15 is 2/3 14 / 21 goes 7 go both 2/3 and 18 or 27 9 goes into both and that would be 2/3 also so they each have the same ratio so this one would be yes SSS similarity and triangle e G  those with I cannot believe I did that do not put that congruent son right there talk that out big-time that should just be a similarity statement guys I am so sorry triangle what matches up with e here now he is by the shortest side 15 is the shortest side over here so eat and G would be the same as m2l so if I do EFG I'm going to across the long way in hell so be careful when your name and that you name it in the corresponding order what's the spawn the value of x that makes triangle rst similar to Triangle hgk then when we are doing that if you will look you have an X here and X here so we need to find the two sides that match up that don't have X's so let's look at that in from here we're going to have to get this up here RT is the first two letters in that goes with HG so we know they are I'm sorry art shg  NST or the second two letters that goes with g k St goes with g k and then that would leave the first turn the last star in the T to go with the H&K now look at what we have here and let's fill this in if we look at RS right here that's three  mhg is x + 7 St is 5 in CK is 20 / 3 RT is 3 - x + h k is 8 I'll be equal the hair we are set that up and solve for x here's the deal you're going to pick any two of these you can just pick this one and cross multiply you can pick this one in Cross Supply in the same problem the X should all be the same so just pick one of them I'm going to pick this set and I'm going to solve it using that so what I'm going to do here is imma make this a little bigger so I can see and saw that if I cross multiply when I do three times 20 over real quick 3/1 * 20 / 3 by 3 Skins when I get 20 over what so this cross is going to be 20 equal this cross you have to distribute the 5 so 5 x + 5 x 7 is 35 now just saw it algebraically - 35 on each side  can you get negative 15 equals 5x divided by 5 you get x equals negative 3 now that's going to be your answer for this problem is that X is going to equal -3 now it didn't matter if you would have done these two together still going to get X's -3 when you cross multiply ls8 the shortest side of a triangle similar to triangle ABC is 15 units on find the other side length of the triangle well here's your ABC right here at so let's try another little triangle  and it says the shortest side of the angle is similar to that is similar to ABC is 15 what's the shortest side of your sick so let's label it correspondingly this is 15 and then we know that Medium Suds here in that large side is here because the media matches up at 7 in the night that she's up with large how we can solve this we need to find the medium and we need to find the large well we know these are similar so what does that give me if I can match up one set aside it gives me that so my scale factor here is 6 to 15 now that will reduce by 3 3 will go into 6 X so 2/5 is my scale factor here so I can set up these problems equal to the scale factor right I'm going to set up the medium with 7 and I'm going to set up the large with the 9 I have two problems to solve to find each side so if I use that scale factor 2/5 now that doing this triangle first to this triangle so if I do mediums I got to do the 7 with the medium make it an x 7 / X cross multiply 2X equals 35 divided by 2 and you get 17.5 that's my medium size 17.5 now it's due the large still the 6/15 and the 9 over the large or do the two Fest I'm going to go ahead and match up sides this time just to show you that scale factor or just sides give you the same thing so Medusa 6/59 over my ex so 6 over 15 equals 9 over X Sin City using the scale factor used to side cross multiply 6 x 15 * 9 is 135 / 6 you get 22 point pause small large side is 22.5 and I saw that whole triangle problem on this page number 9 Pichu or has sides do five and six Stu has sides of 2.52 and 3 are the triangles similar will we only know it sides so we would have to use side side side similarity correct so again you play match up the small with the small of meeting with medium large with the large so look here this is my small my medium my large but look here this is the small this is the medium this is the large so I'm going to match up this one with this one so 2/2 is what okay let's do the mediums the mediums would be the live in the 2.55 / 2.5 is not one it's two over one so I don't have the same ratio already that's enough to say no they are not similar  last page last three problems we're going to do some word problems with similarity now these are the kind of problems you turn the four with the ladder against the wall in the Shadow with the tree and all of that kind of stuff so we can use these in the real world to find measure some very tall things that are hard to measure with a tape measure such as a building and such like that so what we do is we draw a triangle that we can measure it and parrot to a triangle that we can't so we can find the unknown now number one says when Brady stands 2 feet from a Lamppost his shadow is three if Brady is 6ft tall how tall is the lamp post okay you got a backup and draw this out Brady stands 2 ft from a Lamppost to make a lamp post of some type and Brady is standing 2 feet from it  his shadow is receipt long know which way is this Shadow going to be if you stand from lamp I was just going to be away from the lamp right so that's 3 ft if bright is 6 ft tall how tall is the lamp post well Shadows are cast and they're going to come down from the same angle so if you notice here you're going to have two triangles you're going to have triangle right here 6 and 3 and then you're going to have the big triangle which is X but this has to be the whole side so that whole side is going to be 5 so it's like you have two little triangles one is 3 and 6 and the other bigger one is 5 Annex so make a ratio proportion out of that you can do 6 / 3 x / 5 you can do 6 / x 3 / 5 just make you stay in the same pattern so if I do the height of Brady over the height of the lamp-post equal Brady's  Shadow over the lamp post Shadow across multiply 3x equals 30 equals chance so the lamp-post is 10 where in feet talk  now we're going to go outside and actually Newsies in the next week or so and you're going to do one with the shadow but you're also going to do this one like the second one like the mirror and it says Bryce measure the height of a building in the mirror that on the ground face up the mirror was 17 ft from the building and price was 2 ft from the mirror if prices eyes were six feet from the ground how tall is the building okay straw this out again the mirror is 17 ft away from the building so if we have a building 17 ft away from that there's a mirror laying Legrand what happens is you like that on the ground and you walk backwards away from it in the opposite direction going this way until you see the tip of the in the mirror and for Bryce he had to go to feet to where he saw with his the tip of the building in that Mary and then the tip of the building with that point in the mirror makes a triangle so here's our two triangles right here and right here now is Bryce's eyes or six feet from the ground he's looking with his eyes to see the mirror so he's not looking with the top of his head so you have to go from his eye height here's our two triangles so if I did the height of rice over his shadow in the height of the building over it's it's not a shadow but distance to the Mirror you've got your ratio proportion to cross multiply so 2X equals 102 divided by 2 and you get 51 that building has to 51 Waring feet tall in the last one number three and a walks away from a tree long it Shadow into her head is in the line with the top Treetop at the end of The Shadow Anna standing 15 feet from the base of the tree in a 6ft from the end of her show who is 5 feet tall what side of the tree go back take it step-by-step and a standing 15 feet from the base of a tree so you have a tree can 15 feet from the base is Anna and six feet from the end of her shed now her Shadow from the Tree of course with the Sun is going at that same angle so basically she walks to where the tip of her shadow in the tip of the tree shadow find up so for her she's six feet from the end of The Shadow now she's 5 ft tall how tall is the tree again make sure that when you do this triangle you add these two together that's 21 so the tree Shadow is 21 and a shadow is 6 so if we do Anna's high over Ana Shadow and the tree height over the tree Shadow we have a ratio proportion cross multiply 6X equals 1 5/6 and you get 17.5 and we are in feet .5 ft so the trees 17.5 feet tall you have a worksheet for this if you have any questions come see me

Uploaded by Pam Chastain.

Videos

8.2 & 8.3 Notes

More videos from Pam Chastain