Notes 4.1 Triangles, Sum, and Exterior Angles

[00:00:00]hey everybody today we're going to talk about notes point 4.1 which is on triangles the triangle sum theorem and the external angle theorem so first of all let's talk about triangles which is something you've probably known since roughly middle school here but a triangle is of course a figure formed with three segments uh by joining the three non-colinear points so now that you're in geometry we're going to talk about how to name a triangle so we name a triangle using its three letters all right and so the example that i have for you in the picture that i have is that we would name this one triangle abc now somebody else could name this triangle cba or bac or maybe cab there's lots of different names and that order of the letters will become important but it's not just yet all right so the next thing we have to talk about is the vertex so uh there are uh three of them and in this triangle we have vertex a vertex b and vertex c so if you haven't figured that out yet that would be the three points of the triangle and again like i said in this one that was vertex a vertex b and vertex c all right and then we also have the line segments of the triangle which are the sides so we have side a b we have side bc we have side ac those are the three sides of the triangle so these are the segments and that would be a b b c and c a or a c however you want to write it okay um now probably one of the most important things that you are going to learn in geometry is the triangle sum theorem it's amazing how much this theorem comes up and how much you're going to use it this course but here it is it says the sum of the measures of the angles of the triangle is 180 degrees all right so if you add up all the interior angles of the triangle if you add up measurement angle a plus measure angle b plus measure angle c you're going to get 180 all right that's going to be what they sum up to so let's practice using this new theorem here and a few examples so our first example here i have this triangle it's a 45 degree angle a 70 degree angle and an x degree angle we want to solve for x so to do this problem i need to remember that new theorem i learned that all the angles x plus 70 plus 45 have to sum up to 180. after i do this it's really just an algebra problem i can combine my like terms and get x plus 115 set it equal to 180 subtract the 115 from both sides and i would end up with 65 degrees all right so that's how how much x is it is 65 degrees uh the other two are going to be for you to practice but i'm going to give you a little hint here this angle needs to be in the problem all right and this angle has a measure to it so what does this symbol mean and how many degrees is that angle that i've highlighted there in blue all right make sure you add up all three angles set it equal to 180 all right let's talk about naming triangles based on their side lengths it's the first triangle that we're going to talk about is going to be called an equilateral triangle and equilateral triangle as you can see my picture there the picture has all equal sides one two three they're all equal it also has all equal angles uh and turns out that when you actually measure those angles all those angles are going to have the same measure they're all going to be 60 degrees in an equilateral triangle because we have to take our 180 divided by uh three and we get six degrees a piece all right let's look at our second one there our second triangle is going to be our isosceles triangle and isosceles triangle it has two equal sides but it also comes with two equal angles so if you see in my picture here i have my two equal sides right there and we have our two equal angles at the bottom there's the base angle that's what we call those okay um so when you look at this problem here um we really get uh two equal things so look at what we have we have we have something with three equal sides so we have two equal sides now you can't you can't have something with one equal side doesn't make sense right so you're gonna have a triangle with no equal sides all right and that's going to be our last one here this is going to be a scaling triangle so a scalene triangle as i have there in the definition is a triangle with no equal sides and but it also turns out that it has no equal angles as well all right and that's how we get our scaling triangle you can see my picture there all the side lengths are different that one's six this one's twelve this one's eight all right so they're all different side lengths therefore this is the scaling triangle all right let's practice let's try to identify a couple triangles in this picture here so to determine whether it is uh equilateral isosceles or scaling triangle the first problem we're going to do together is triangle bed i'm going to outline it in blue so i'm going from b to e study and of course back to b and i get this triangle in blue gear now because we're identifying as equilateral isosceles or scalene i need to identify based on the sides so i'm going to look at the side lengths and see if they're marked this triangle has two markings on this side and two markings on this side and a single marking on its third side so what i can take away from that is that there are two equal sides in this triangle which means it is an isosceles triangle all right let's do one more together here let's grab a different color here and let's go to triangle d a b so we're going from d to a to b going triangle d a b and purple here and again let's look at the markings on the three sides of my triangle here there's a one mark there one right there and one mark there and so because each one of them has one marks there are three equal sides in this triangle which means this is an equilateral triangle all right your last two are for you to try you'll find them in your quick check but they work exactly the same as i've done here let's talk about something else let's talk about naming triangles by their angles so we're going to have three different names here as well so the first one we're dividing as all angles all three angles the triangle are less than 90.

[00:05:53]now do you know any other names for angles that are less than 90 you probably do you probably know it's called an acute angle when we have three acute angles in a triangle we call that an acute triangle all right you notice all my angles there are less than 90.

[00:06:10]all right the second one this one's defined as one right angle in the triangle or one angle that equals nine degrees and uh you probably know a 90 degree angle it's called a right angle so this is called a right triangle probably know some other things about right triangles but we'll cover those again later this year and the last one we have is one that is defined by one angle that is greater than 90 or one angle that is obtuse right and an obtuse angle leads us to an obtuse triangle all right so there are three classifications based off of the angles of the triangle it's a pair with our three classifications based on the sides of the triangle so now we can double classify our triangles which is exactly what i'm about to do with you now let's look at example five it says to classify each of them by their angles and by their sides so we have to kind of keep both classifications in mind here so let's go into this let's go ahead and classify based on the angle first i see the right angle so i'm going to lead into a right triangle but i also need to pair it with the name of its side classification and so i look at the triangle and i see it has two equal sides beam marks that's going to be an isosceles so we have a right isosceles triangle right also helps if you spell isosceles right sometimes it's a little tricky right i s o s there we go c e l yes all right and let's do one more here let's go with uh example c and so when you look at this one you're looking at those angles you're like ah all these angles are pretty small now they're more than 90 now they're very large so this must be an acute triangle here and then i'm looking at the sides and i see one marking here and a double mark in there so those sides aren't equal the third side's not marked so that must be a case where none of the sides are equal aka a scalene triangle so we have an acute scalene triangle all right parts b and parts d are very similar to this one those are for you to try all right our last theorem of the day is the exterior angle theorem this theorem says that the measure of the exterior of the angle is equal to the sum of the two remote interior angles so with this problem what we want to do is first of all identify what is the remote interior angle so i'll put it right over here we have remote interior angle and that's actually going to be my angle a my angle b so those are my remote interior angles and you probably can identify then that this was the exterior angle because you're familiar with the word exterior probably meaning outside and this angle is outside of the triangle all right so it turns out that when you add up the two measures of those remote interior angles so you add up the measure of angle a and you add up the measurement a plus a measure angle b it actually equals the measure of the exterior angle which is the measure of angle one right so you can do that on any of the triangles you have an exterior angle by the way they're called remote interior angles because they are farthest away from the exterior angles so you never want to pick this one this one is not next to uh that one's like directly next to your exterior angle so it's not remote from it all right so that's kind of your comparison there all right let's do a few examples down here um so in our first example here we have the problem where you have x a 55 and a 125 and basically what you need to understand is that you have the 125 is your remote angle so i'm gonna go ahead and start off with that one i'm gonna say 125.

[00:09:23]sorry that's the exterior angle excuse me on my wording sorry that is your exterior angle all right your remote angles remove interior angles are these two right here the x and 55 and those two are going to add up with each other and sum to the exterior which is 125. it's going to be a pretty simple algebra problem here we're just going to subtract 35 on both sides and once we do we find out that x is equal to 70.

[00:09:44]all right i'm going to go ahead and help you start the next problem but i am going to save these for you to finish all right let's look at number seven uh in number seven we have our exterior angle being our 2x plus 10 and the remote interiors are the x and the 65 there and so i'm going to go ahead and have that 2x plus 10. i always start with the outside one always over the outside one set it equal to the other two added together so you're gonna set that equal to x plus 65 you're going to solve that one using your different algebra techniques and find x all right example 8 will be for you to try in your quick check completely by yourself and then we have one last thing to talk about down here which is that i have equilateral you know i have this double arrow thing and then equal angular right so it says that if all three sides of a triangle are equal then all three angles will be equal and all of the measures of the angle will be 60. so what it's trying to say uh is that basically if you have an equilateral triangle you have an equal angular triangle and vice versa and then if you have equal angular you also have equilateral they come hand in hand that's all it is so if you ever see a case where you have all equal sides you know the angles are equal and you know the exact opposite of that is true as well all right that wraps up for 4.1 if you guys have any questions feel free to let us know