unit four section one classifying triangles we moved on from just a point one point like one spot in space to then lines where we connect a bunch of points together and then we did planes where we connect a bunch of lines together now we're going to connect three lines together that's where we get the tri in triangles try meaning three okay so if you want to write that down that's probably good try equals three and it's a try angle you could even call it a tri-cider if you wanted to because it will also have three sides when it has three angles we're going to be able to classify a triangle by its angles and its sides if it has three acute angles it is an acute triangle wrote that all kinds wrong here we go acute ends with an e and a triangle looks like this it's an acute triangle okay it might have one obtuse angle if it has one obtuse angle then it is an obtuse triangle it's going to have a really long what we call the hypotenuse okay a if it has one right angle then it is a right triangle and if it has three congruent angles congruent means same or equal that is marked by these little arches right here in there that means that all three of those angles are the same measure then that one is going to be an equiangular equingular now if we break that word down equa means equal angular means angle so that means equal angles okay so that's what it is so we're gonna use the diagram classify each triangle by its angles so let's give it a go if we look at number one it says triangle c d e so you go from c to back to e back to c so we're looking at this one and what i see immediately is in that right hand corner it's got that box that box that box is 90 degrees okay so that means that that box is 90 degrees that's a right angle so that means that is a right triangle and it really helps to draw on top of these things and then erase so pencil is a good thing to have bce start at b go to c go to e this triangle right here we've got this big old angle that's an ob2 so obtuse triangle and i like putting the symbol for triangle instead of right and triangle you're gonna have to say the word triangle and right triangle a bunch just use the symbol so just make a triangle for triangle if you want make an angle for angle i don't like writing them over and over a b e a to b to e boom bam bam bam all three angles are the same all three angles are equal equal angular triangle now you don't have to capitalize the a and equal angular i am so that you can see that it's basically two little pieces of a word stuck together to make one whole thing equal equal angular angle equal angle b d e from b to d to e b d e that guy right there big ol angle we're going to call that one obtuse it's an obtuse triangle and then we got c f e c to f to e c f e now that box is there but there's this little wedge on the outside that goes with that box so that's not 90. this one's less than 90. that one's less than 90. that one's less than 90. they're all small they're not all the same because they don't have the little arcs in them so that would be an acute triangle okay cute triangle that's what that's like you can also classify them by sides if they have three congruent sides all the sides are the same or equal in measure okay we're going to call that equa we use that word already lateral okay equilateral triangle equa of course means same or equal lateral means side okay like football if you do a lateral play you throw the ball to your side you throw it laterally okay that's what that means just talking about which way so equilateral all the sides are equal equilateral triangle is also going to have equal angles you can call it either name equilateral or equal angle if it is equilateral it is equal angle if it is equal angle it is equilateral because the sides of a triangle are directly tied and in reference to the angle that they have because these two sides are going to form the third side by that angle so this length is determined by this angle and this angle makes that length and this angle makes that length then if they're all the same then they're all the same if it has two congruent sides so one long piece and two congruent sides that is called isosceles okay it's an isosceles triangle and another thing you're going to know about isosceles triangle we just talked about it in the last one if this side is the same as that side and they we know that because they're marked then the angles opposite them that made them are also the same so if it has two congruent sides it's isosceles but if it has two congruent angles it is also isosceles if it has no congruent sides we call that scalene and it's just a word you're gonna have to memorize it's a scalene triangle any triangle that's not isosceles or equilateral scalene it's going to have three different amounts of sides or three different amounts of angles because this one is not the same as this one which is not the same as that one they're all different okay so let's use the diagram and classify each triangle by its sides let's look at j to k all the way over to m this guy right here boom boom i see one has a mark but it's not the same as km and it's not the same as jm and they're not the same as each other none of them are the same so that one is going to be scalene now make sure you realize that it does say by side so when we use the equa we should use the equilateral not the equal angular even though you can use them interchangeably let's look at klm from k to l to n these two sides are the same it's got two sides that are congruent so that's isosceles hello if i can spill out sauce loser and put the l first e l e s triangle now we've got three choices and we've got three problems and we've used two so most likely it's going to be equilateral but let's just check to make sure j k n from j to k to n one two three sides the same equilateral triangle very good okay now depending upon sides and angles you should be able to identify what kind of triangle this is now if we're doing it by angles remember if we go back a page we've got different kinds of angle triangles acute obtuse right equal angular and if we're looking at sides we have equilateral isosceles and scalene it's asking us to do both this and means both things so number nine i see a 90 degree angle in there so i know it is a right something because it has the 90 degrees right there that's 90 degrees it has two sides that are the same so it's isosceles if i could spell isosceles right twice in a row e-l-e-s there we go it's a right isosceles triangle okay now if we look at number 10 what we know about number 10 is it's got one big old angle in there so we know it's obtuse it doesn't tell us anything about the other two angles or any of the sides so we just assume they're all different so that means it's scalene number eleven on the inside i see sixty sixty sixty all the angles are the same and i also see that all of the sides are seven seven seven so it is an equa angular q-u-i equilateral triangle usually you don't say both of those at the same time you say one or the other because if you know one you know the other if all the angles are the same all the sides are the same the side opposite the angle has a connection the angle creates that side the smaller angle is going to be the smaller side the middle angle is going to be the middle side and the biggest angle is going to be the bigger side and any that are the same are going to be the same number 12 all those sides are different it doesn't tell us anything about the angles but if all the sides are different that means all the angles are different that means that they are scalene and then what we have to determine is well what kind of angliness is it well there's not an obtuse angle in there i see they're all pretty small if i mean i can make a line here for 90. i can make a uh let's see where would i i would do like there's an angle there for 90. there's an angle there for 90. none of them are 90 they're all smaller than 90. so we would call that an acute scalene triangle 13. i got that 90 right there that's right it doesn't tell me anything else about any of the other sides so i assume they're all different and less marked so it's a right scalene triangle 14 97 is big so we know that's obtuse anything over 90s obtuse we have a five and a five those are the same two the same sides means isosceles let's see if i can spell right this type look at that i can okay so that is going to be an isosceles right or an obtuse isosceles triangle all right if p r bisects s r t p r which one's pr pr goes this way if it bisects cut that's the sect part two is two cut so it's going to cut it into two exactly so we know that this side is equal to that side oh sorry nope that line is bisecting s r t if p r the angle srt so s to r to t so this angle bisects so we know this angle and this angle are the same so i could put a little 30 in there because i know that's 30 because bisect means cut into exactly in half let's see point u is the midpoint of rt so point u is in the middle so if this that are the same and that is the same because it was marked on r uq triangle so we know all three of those guys are the same classify each triangle by its angles and sides okay no problem u q r from u to q to r well that 120 is obtuse and we've got two sides that are the same so it's isosceles there you go i'll do 16 with you and then i'll let you try 17 18 19 20. yet at all times you can always just pause right before i start try the problem yourself and then see how i do it uh r s t r s t okay well eight plus eight is sixteen and this side is sixteen and thirty and thirty is sixty and sixty sixty sixty so that's equal angular equilateral angular and equilateral all right go ahead and pause right now if you haven't been pausing already and try 17 18 19 and 20 and then come back and see how they go all right let's look at 17 srq from s to r to q it's got a 90 so we know it's right 30 60 90 they're all different so it's going to be scalene p r t from p to r to t that's 16 this is eight so that makes that eight eight and eight makes 16 so that's isosceles for sure that's 60. whatever that is it makes more than 90 for sure so it's gonna be an obtuse right i know isosceles triangle t q u from t to q to u oh interesting well i learned that this was eight and that one's eight so these are all the same they all have the same sides that means that they all have to have the same angles i can put those in there plus this is a line and that cuts with 120 and 180 minus 120 is 60 so i know that that's 60 and then uh 60 plus 60 makes 120 and the other part of that to 180 i just figured it out was 60 so all three are 60 so it's not only equiangular but also equilateral pqt from p to q to t well that one's eight that one's 16 so they're not the same so it's going to be scalene if this one's 120 that one's 30 those sides are the same that one has to be 30 60 and 30 makes 90 so that means that this has to be 90 plus those are vertical angles this way to that way or vertical so that's 90. so it is a right scalene triangle awesome now let's throw some algebra in there if abc is equilateral that means this side and that side and that side are all equal solve for both x and y well i see one part with an x and i see one that's just a number and i know that they're equal equal so i can say 3x plus 1 equals 49 and i can solve for x minus 1. 3x equals 48 divided by 3. let's see where's my calculator now we'll use that one so we'll do 48 divided by 3. perfect exactly what i thought it was 16. x equals 16. look at me showing work i got another one with y and i know that it's 49 so i can say 49 equals 18y minus 41. add 41 to both sides 0 carry 8 9. divide by 18 y equals let's see 9 goes into 18 twice 9 goes into 90 10 times so that should be 5. well let's just double check 90 divided by 18 is 5. very good and there's my puff that's how you do those you figure out the connection they tell you in the clue word in the instructions or you can see like if they mark it so try number two give it a pause try number two okay here we go rst is equilateral so we know this side and that side and that side are all the same and they're all x find x and the measure of each side so i need to know how long each side is so not only do i have to find x but i also have to plug it in well since they're equal what i can do very quickly is i can just set one side equal to the other minus 10x that makes 2x minus 22 equals negative 6 at 22. 2x equals let's see 22 minus 6 take off 2 leaves you at 20 and 4 take it down 16 divided by 2 x equals 8. and if x equals eight you'll plug it in easy multiply is eight times ten so i'm gonna use this side eight times ten is eighty minus six is seventy four for the sides 74 side measure now let's double check 56 plus 18. let's see that'd be 66 plus 8 would be 70 and then 4 more would be 74. yeah that works and then the 12 times 8 is 96 minus 2 is 94 minus 20 is 74. uh if i plug it into all three it all works so 74 is the side length and number three if you want to give it a pause give it a try and then come back let's see how it goes all right d e f is isosceles so two sides are the same i'm going to pick the ones that look the same with d e and e f they even tell you which ones are the same find x and the measure of each side so we got to do two things find x and then tally it up i can't use the bottom x the reason i can't use x plus 21 is it's not equal to anything but i do know that 8x minus 13 should be the same as 5x plus 17 minus 5x both sides so 3x minus 13 equals 17. add 13. three x equals thirty divide by three x equals ten and then i'm going to plug it in but i am going to plug it in the bottom because ten plus twenty one equals thirty one now i should check my work five times uh 10 makes 50 plus 17 is 67 i also need to know that sign so 67 this one is 31. this side over here should be 67 80 8 times 10 is 80 minus 13 80 minus 10 is 70. 70 minus 3 is 67 boom so make sure you get both sets of side lengths on there and that is classifying triangles and then using those classified classifications for algebra