Notes 4.2 Isosceles and Equilateral Triangles

[00:00:00]hey everybody today we're going to talk about nuts 4.2 which is on equilateral and isosceles triangles and some of the properties that they go into so our first one that we're going to talk about is going to be the equilateral triangle um right off the bat here i have a picture of an equilateral triangle and it says that if a triangle is equilateral then it is also equal angular so that would mean that if we have an equilateral triangle with all equal sides then we can also assume that all the angles are equal so the measure of angle a would be equal to the measure of angle b which would be equal to the measure of length of c and we can actually determine these angles measures they're all going to be the same because we have 180 degrees divided three ways equally would make 60 degrees a piece so we're going to have a 60 degree angle in every single one of the angles of an equilateral triangle the second part is that if it is equal angular then it is also equilateral so this goes hand in hand like back and forth either way all right so here you can see i have an equi angular triangle all my angles are equal and that just means that every single side is equal so a b would equal to bc which would be equal to ca they're all equal here all right so that's going to be our first property that we're going to demonstrate in a couple examples let's go ahead and do those now first example we want to find the value of x and what we have there is we have an equilateral triangle as we can see marked by all the sides and so that tells us that every angle is going to be 60 degrees we have 60 60 and 60. now you'll notice down here in this bottom right hand corner i have a 60 and i have a 5x so there's two things representing one location which means these two things must be equal so 5x is e 5x is equal to 60 and then when we divide by 5 on both sides we quickly find out that x is 12. all right our second problem is we have an equi angular triangle we notice because we have all equal angles so that tells us that the triangle is going to be equilateral as well so all of my sides are equal or in other words x has to be 17 because all the sides have to be 17.

[00:01:43]our last problem is an equivalence triangle as well and you'll notice this one's a little bit more algebraic we have these two sides that are given as 2x plus 5 and 5x minus 1 but because all the angles are equal we have an equilateral triangle as well as an equal angular triangle so all these sides are equal so i can set 2x plus 5 equal to 5x minus 1 and then i can solve it so we're going to go ahead and subtract 2x from both sides and end up with three x minus one i'm going to add one back and get three x is equal to six and divide by three to find out that x is equal to two you'll find that examples four and five are very similar to the ones i've completed here so you'll find those problems in your quick check let's continue on and talk about some properties of isosceles triangles so right off the bat here we have that if two sides of a triangle are congruent aka it's isosceles then the opposite angles are also congruent so what you can do is you can follow these equal sides you just kind of crisscross them like this and then you'll find of course these angles that i have highlighted over here in blue so that would mean that in this case when a b is equal to bc then we can conclude that angle a is congruent to angle c well it turns out this theorem actually works forwards and backwards as well because our second part of it says if two angles of a triangle are congruent aka angle a and angle c are going to be congruent then the opposite sides are congruent so you can follow them just kind of crisscross them to their opposite sides to find a b and b c are going to be equal here which i've represented in my second drawing as well let's see this property work in a couple examples so our first example here number six um we have the x and technically we started off the problem with the 59 and i kind of left this extra one in here but what you're going to do is you're going to use the property that we have an isosceles triangle so there's two equal sides to carry this angle 59 degrees and match it up over there because these two have to be equal once we do that we now have all three angles of the triangle and i can add up x plus 59 plus another 59 and set it equal to 180 i can then follow my algebra rules and go ahead and solve this problem so i have x plus 118 equals 180 and subtracting 118 from both sides i find out that x is equal to 62.

[00:03:42]i'm going to help you set up the next couple problems but i'm going to save them for you to complete so number seven what we want to do with this one is actually very similar number six you notice you have these two isos or these two equal sides which tells us it's an isosceles triangle um and then what we can do is we can use the property where we can then find two equal angles and all we have to do is just have to go crisscross just like this and you'll find out these two angles are going to be the same so one of them is x degrees so the other one must be x degrees as well again don't forget the fact that all triangles must add to 180 and you can set this problem up now you have all three angles x plus x plus 44 is equal to 180.

[00:04:17]you'll find the third example works very similar um just make sure you're following this property where you crisscross the angles and find them congruent let's do a couple more examples on the other side example 9 i always tell my students is the easiest hard problem you ever do all right or the hardest easy problem maybe i said that backwards but this one it confuses a lot of students and it's very simple you notice you have two congruent angles so that means that we must have two congruent sides simply follow them across the opposite side and you'll find the two that are congruent notice i'm pointing at a six and i'm putting in an x so that must mean that x is equal to six and i'm done example number ten uh again similar property we're going to follow these angles across to the other side two congruent angles so we must have two congruent sides must be these two and i can say two x plus four is equal to three x minus ten and i can finish that by solving it algebraically which i'm gonna save for you next example is also in your quick check cause you'll find number eleven completed on your own be careful on this one notice there is all three sides of foot out here so there is gonna be some extra information that you won't need um but just kind of take that as a hint let's look at example number 12.

[00:05:19]in example number 12 we have uh we notice we have the exterior angles from the from the last lesson from 4.1 included in this one as well so we're going to kind of combine our two topics what we need to do is start off with the fact that we have these two angles or these two sides excuse me being congruent and so that can go ahead and tell us that we can eventually find out that their angles are also going to be congruent so i can figure out these two are going to be the same but the other fact that you can really use to help you hear is that right here this is a straight line this purple line is a straight line right so if i can determine what this little spot right there is well then i can determine what y is right so i can say 123 plus the unknown because it's a straight line has to be 180 degrees so if i subtract 123 from both sides i quickly find out that the unknown is gonna be 57 so because i know this spot here is 57 i now know that y is equal to 57 as well by my new property today so this is going to be 57 degrees and then how do i find x well i have all three angles of my triangle now so now i can kind of add them all up and set it equal to 180.

[00:06:21]so we can say 57 plus 57 plus x is going to equal to 180.

[00:06:27]if i solve this problem i'm going to find x so i'm going to go ahead and add and get x plus 114 is equal to 180 and once i subtract 114 from both sides i found out that x is 66. you'll find example 13 very similar to number 12.

[00:06:42]so that one is for you to complete let's go do one more example on the back page uh number 14 it is very similar to the one that we just worked but it works in a little bit different order so when you look at this problem here we have an x we have a y it has 6 2 they're in different spots than before so in this problem what you want to do is you actually want to move the y over just like we did last time because again you can kind of find these two congruent angles so we can call this y degrees right there and then i can focus in on the triangle itself i can focus in on this because this part i have all the angles of this at least i have all the angles filled out and so then i can actually solve this so i get y plus y plus 62 is equal to 180.

[00:07:17]so the reason i can do this one is because my equation is in just in terms of y so it's actually very easy for me to solve for combining the y's i get 2y plus 62 is equal to 180 and i can subtract 62 from both sides to get to 118.

[00:07:32]whoops not a thousand there about divided by two and we find out that y is equal to 59.

[00:07:38]okay well again we can go back we can fill in those 59s if we want and there's actually two different ways to get the x i'm going to use the one that i just used in the last problem which is again straight line right here and so if i have a straight line i can go ahead and say this x plus this 59 is going to equal to 180 because it's a straight line and i can go ahead and subtract the 59 from both sides and find out that x is equal to 121. the other way would have been to use your x to your angle theorem from the last problem you could have added 62 and 59 and also got 121. that's it for this one guys example 15 is in your quick check so you guys have a good one we'll see ya