Video Transcript: Triangle Angles

Angles in a triangle sum to 180 degree proof 

I've drawn an arbitrary triangle right over here. And I've labeled the measures of the interior  angles. The measure of this angle is x. This one's y. This one is z. And what I want to prove is  that the sum of the measures of the interior angles of a triangle, that x plus y plus z is equal to  180 degrees. And the way that I'm going to do it is using our knowledge of parallel lines, or  transversals of parallel lines, and corresponding angles. And to do that, I'm going to extend  each of these sides of the triangle, which right now are line segments, but extend them into  lines. So this side down here, if I keep going on and on forever in the same directions, then now  all of a sudden I have an orange line. And what I want to do is construct another line that is  parallel to the orange line that goes through this vertex of the triangle right over here. And I  can always do that. I could just start from this point, and go in the same direction as this line,  and I will never intersect. I'm not getting any closer or further away from that line. So I'm never  going to intersect that line. So these two lines right over here are parallel. This is parallel to  that. Now I'm going to go to the other two sides of my original triangle and extend them into  lines. So I'm going to extend this one into a line. So, do that as neatly as I can. So I'm going to  extend that into a line. And you see that this is clearly a transversal of these two parallel lines.  Now if we have a transversal here of two parallel lines, then we must have some corresponding  angles. And we see that this angle is formed when the transversal intersects the bottom orange line. Well what's the corresponding angle when the transversal intersects this top blue line?  What's the angle on the top right of the intersection? Angle on the top right of the intersection  must also be x. The other thing that pops out at you, is there's another vertical angle with x,  another angle that must be equivalent. On the opposite side of this intersection, you have this  angle right over here. These two angles are vertical. So if this has measure x, then this one must have measure x as well. Let's do the same thing with the last side of the triangle that we have  not extended into a line yet. So let's do that. So if we take this one. So we just keep going. So it  becomes a line. So now it becomes a transversal of the two parallel lines just like the magenta  line did. And we say, hey look this angle y right over here, this angle is formed from the  intersection of the transversal on the bottom parallel line. What angle to correspond to up  here? Well this is kind of on the left side of the intersection. It corresponds to this angle right  over here, where the green line, the green transversal intersects the blue parallel line. Well  what angle is vertical to it? Well, this angle. So this is going to have measure y as well. So now  we're really at the home stretch of our proof because we will see that the measure-- we have  this angle and this angle. This has measure angle x. This has measure z. They're both adjacent  angles. If we take the two outer rays that form the angle, and we think about this angle right  over here, what's this measure of this wide angle right over there? Well, it's going to be x plus z.  And that angle is supplementary to this angle right over here that has measure y. So the  measure of x-- the measure of this wide angle, which is x plus z, plus the measure of this  magenta angle, which is y, must be equal to 180 degrees because these two angles are  supplementary. So x-- so the measure of the wide angle, x plus z, plus the measure of the  magenta angle, which is supplementary to the wide angle, it must be equal to 180 degrees  because they are supplementary. Well we could just reorder this if we want to put in  alphabetical order. But we've just completed our proof. The measure of the interior angles of  the triangle, x plus z plus y. We could write this as x plus y plus z if the lack of alphabetical order 

is making you uncomfortable. We could just rewrite this as x plus y plus z is equal to 180  degrees. And we are done. 

Proofs concerning isosceles triangles 

So we're starting off with triangle ABC here. And we see from the drawing that we already  know that the length of AB is equal to the length of AC, or line segment AB is congruent to line  segment AC. And since this is a triangle and two sides of this triangle are congruent, or they  have the same length, we can say that this is an isosceles triangle. Isosceles triangle, one of the  hardest words for me to spell. I think I got it right. And that just means that two of the sides are  equal to each other. Now what I want to do in this video is show what I want to prove. So what I  want to prove here is that these two-- and they're sometimes referred to as base angles, these  angles that are between one of the sides, and the side that isn't necessarily equal to it, and the  other side that is equal and the side that's not equal to it. I want to show that they're  congruent. So I want to prove that angle ABC, I want to prove that that is congruent to angle  ACB. And so for an isosceles triangle, those two angles are often called base angles. And this  might be called the vertex angle over here. And these are often called the sides or the legs of  the isosceles triangle. And these are-- obviously they're sides. These are the legs of the  isosceles triangle and this one down here, that isn't necessarily the same as the other two, you  would call the base. So let's see if we can prove that. So there's not a lot of information here,  just that these two sides are equal. But we have, in our toolkit, a lot that we know about  triangle congruency. So maybe we can construct two triangles here that are congruent. And  then we can use that information to figure out whether this angle is congruent to that angle  there. And the first step, if we're going to use triangle congruency, is to actually construct two  triangles. So one way to construct two triangles is let's set up another point right over here.  Let's set up another point D. And let's just say that D is the midpoint of B and C. So it's the  midpoint. So the distance from B to D is going to be the same thing as the distance-- let me do  a double slash here to show you it's not the same as that distance. So the distance from B to D  is going to be the same thing as the distance from D to C. And obviously, between any two  points, you have a midpoint. And so let me draw segment AD. And what's useful about that is  that we have now constructed two triangles. And what's even cooler is that triangle ABD and  triangle ACD, they have this side is congruent, this side is congruent, and they actually share  this side right over here. So we know that triangle ABD we know that it is congruent to triangle  ACD. And we know it because of SSS, side-side-side. You have two triangles that have three  sides that are congruent, or they have the same length. Then the two triangles are congruent.  And what's useful about that is if these two triangles are congruent, then their corresponding  angles are congruent. And so we've actually now proved our result. Because the corresponding  angle to ABC in this triangle is angle ACD in this triangle right over here. So that we then know  that angle ABC is congruent to angle ACB. So that's a pretty neat result. If you have an isosceles triangle, a triangle where two of the sides are congruent, then their base angles, these base  angles, are also going to be congruent. Now let's think about it the other way. Can we make the other statement? If the base angles are congruent, do we know that these two legs are going to be congruent? So let's try to construct a triangle and see if we can prove it the other way. So I'll  do another triangle right over here. Let me draw another one just like that. That's not that 

pretty of a triangle, so let me draw it a little nicer. I'm going to draw it like this. Let me do that  in a different color. So I'll call that A. I will call this B. I will call that C right over there. And now  we're going to start off with the idea that this angle, angle ABC, is congruent to angle ACB. So  they have the same exact measure. And what we want to do in this case-- we want to prove--  

so let me draw a little line here to show that we're doing a different idea. Here we're saying if  these two sides are the same, then the base angles are going to be the same. We've proved  that. Now let's go the other way. If the base angles are the same, do we know that the two  sides are the same? So we want to prove that segment AC is congruent to AB. Or you could say  that the length of segment AC, which we would denote that way, is equal to the length of  segment AB. These are essentially equivalent statements. So let's see. Once again in our  toolkit, we have our congruency theorems. But in order to apply them, you really do need to  have two triangles. So let's construct two triangles here. And this time, instead of defining  another point as the midpoint, I'm going to define D this time as the point that if I were to go  straight up, the point that is essentially-- if you view BC as straight horizontal, the point that  goes straight down from A. And the reason why I say that is there's some point-- you could call  it an altitude-- that intersects BC at a right angle. And there will definitely be some point like  that. And so if it's a right angle on that side, if that's 90 degrees, then we know that this is 90  degrees as well. Now, what's interesting about this? And let me write this down. So I've  constructed AD such that AD is perpendicular to BC. And you can always construct an altitude.  Essentially, you just have to make BC lie flat on the ground. And then you just have to drop  something from A, and that will give you point D. You can always do that with a triangle like  this. So what does this give us? So over here, we have an angle, an angle, and then a side in  common. And over here, you have an angle that corresponds to that angle, an angle that  corresponds to this angle, and the same side in common. And so we know that these triangles  are congruent by AAS, angle-angle-side, which we've shown is a valid congruent postulate. So  we can say now that triangle ABD is congruent to triangle ACD. And we know that by angle angle-side. This angle and this angle and this side. This angle and this angle and this side. And  once we know these two triangles are congruent, we know that every corresponding angle or  side of the two triangles are also going to be congruent. So then we know that AB is a  corresponding side to AC. So these two sides must be congruent. And so you get AB is going to  be congruent to AC, and that's because these are congruent triangles. And we've proven what  we wanted to show. If the base angles are equal, then the two legs are going to be equal. If the  two legs are equal, then the base angles are equal. It's a very, very, very useful tool in geometry. And in case you're curious, for this specific isosceles triangle, over here we set up D so it was  the midpoint. Over here we set up D so it was directly below A. We didn't say whether it was the midpoint. But here, we can actually show that it is the midpoint just as a little bit of a bonus  result, because we know that since these two triangles are congruent, BD is going to be  congruent to DC because they are the corresponding sides. So it actually turns out that point D  for an isosceles triangle, not only is it the midpoint but it is the place where, it is the point at  which AD-- or we could say that AD is a perpendicular bisector of BC. So not only is AD  perpendicular to BC, but it bisects it. That D is the midpoint of that entire base. 

Proofs concerning equilateral triangles

What we've got over here is a triangle where all three sides have the same length, or all three  sides are congruent to each other. And a triangle like this we call equilateral. This is an  equilateral triangle. Now what I want to do is prove that if all three sides are the same, then we  know that all three angles are going to have the same measure. So let's think how we can do  this. Well, first of all, we could just look at-- we know that AB is equal to AC. So let's just pretend that we don't even know that this also happens to be equal to BC. And we know for isosceles  triangles, if two legs have the same length, then the base angles have the same length. So let's  write this down. We know that angle ABC is going to be congruent to angle ACD. So let me  write this down. We know angle ABC is congruent to angle ACB. So maybe this is my statement right over here. And then we have reason. And the reason here, and I'll write it in just kind of  shorthand, is that they're base angles of, I guess you could say an isosceles. Because we know  that this side is equal to that side. And obviously, this is an equilateral. All of the sides are equal. But the fact that these two legs are equal so that the base angles are equal. So we say two legs  equal imply base angles are going to be equal. And that just comes from what we actually did in the last video with isosceles triangles. But we can also view this triangle the other way. We  could also say that maybe this angle over here is the vertex angle, and maybe these two are the base angles. Because then you have a situation where this side and this side are congruent to  each other. And then that angle and that angle are going to the base angles. So you could say  angle CAB is going to be congruent to angle ABC, really for the same reason. We're now looking at different legs here and different base angles. This would now be the base in this example.  You can imagine turning an isosceles triangle on its side. But it's the exact same logic. So let's  just review what I talked about. These two sides are equal, which imply these two base angles  are equal. These two sides being equal implied these two base angles are equal. Well, if ABC is  congruent to ACD and is congruent to CAB, then all of these angles are congruent to each  other. So then we get angle ABC is congruent to angle ACB, which is congruent to angle CAB.  And that pretty much gives us all of the angles. So if you have an equilateral triangle, it's  actually an equiangular triangle as well. All of the angles are going to be the same. And you  actually know what that measure is. If you have three things that are the same-- so let's call  that x, x, x-- and they add up to 180, you get x plus x plus x is equal to 180, or 3x is equal to 180.  Divide both sides by 3, you get x is equal to 60 degrees. So in an equilateral triangle, not only  are they all the same angles, but they're all equal to exactly-- they're all 60 degree angles. Now  let's think about it the other way around. Let's say I have a triangle. Let's say we've got  ourselves a triangle where all of the angles are the same. So let's say that's point X, point Y, and  point Z. And we know that all the angles are the same. So we know that this angle is congruent  to this angle is congruent to that angle. So what we showed in the last video on isosceles  triangles is that if two base angles are the same, then the corresponding legs are also going to  be the same. So we know, for example, that YX is congruent to YZ. And we know that because  the base angles are congruent. Now we also know that YZ-- so I'll rewrite YZ-- is congruent to  XZ, by the same argument. But here we're dealing with different base angles. So now, once  again, you can view this as almost an isosceles triangle turned on its side. This is the vertex  angle right over here. These are the two base angles. This would be the base now. And we know that because these two base angles are congruent. So by the same logic. Over in this first case,  the base angles were this angle and that angle. In the second case, the base angles are that 

angle and that angle. And actually let me write it down. The base angles in this first case-- let  me do that same magenta-- are angle YXZ is congruent to angle YZX. That was in the first case.  These are the base angles. So based on the proof we saw in the last video, that implies these  sides are congruent. Here, we have these two base angles. Let me do that in green. Angle XYZ  is congruent to angle YXZ. And so that implies that these two guys right over here are  congruent. Well, there we've proved it. We've said that this side YX is congruent to YZ. And  we've shown that YZ is congruent to XZ. So all of the sides are congruent to each other. So once again, if you have all the angles equal, and they're going to have to be 60 degrees, then you  know that all of the sides are going to be equal as well. They're going to be congruent. 

Triangle exterior angle example 

What I want to do now is just a series of problems that really make sure that we know what  we're doing with parallel lines and triangles and all the rest. And what we have right here is a  fairly classic problem. And what I want to do is I want to figure out, just given the information  here-- so obviously I have a triangle here. I have another triangle over here. We were given  some of the angles inside of these triangles. Given the information over here, I want to figure  out what the measure of this angle is right over there. I need to figure out what that question  mark is. And so you might want to give a go at it just knowing what you know about the sums  of the measures of the angles inside of a triangle, and maybe a little bit of what you know  about supplementary angles. So you might want to pause it and give it a try yourself because  I'm about to give you the solution. So the first thing you might say-- and this is a general way to think about a lot of these problems where they give you some angles and you have to figure  out some other angles based on the sum of angles and a triangle equaling 180, or this one  doesn't have parallel lines on it. But you might see some with parallel lines and supplementary  lines and complementary lines-- is to just fill in everything that you can figure out, and one way  or another, you probably would be able to figure out what this question mark is. So the first  thing that kind of pops out to me is we have one triangle right over here. We have this triangle  on the left. And on this triangle on the left, we're given 2 of the angles. And if you have 2 of the  angles in a triangle, you can always figure out the third angle because they're going to add up  to 180 degrees. So if you call that x, we know that x plus 50 plus 64 is going to be equal to 180  degrees. Or we could say, x plus, what is this, 114. X plus 114 is equal to 180 degrees. We could  subtract 114 from both sides of this equation, and we get x is equal to 180 minus 114. So 80  minus 14. 80 minus 10 would be 70, minus another 4 is 66. So x is 66 degrees. Now, if x is 66  degrees, I think you might find that there's another angle that's not too hard to figure out. So  let me write it like this. Let me write x is equal to 66 degrees. Well if we know this angle right  over here, if we know the measure of this angle is 66 degrees, we know that that angle is  supplementary with this angle right over here. Their outer sides form a straight angle, and they  are adjacent. So if we call this angle right over here, y, we know that y plus x is going to be  equal to 180 degrees. And we know x is equal to 66 degrees. So this is 66. And so we can  subtract 66 from both sides, and we get y is equal to-- these cancel out-- 180 minus 66 is 114.  And that number might look a little familiar to you. Notice, this 114 was the exact same sum of  these 2 angles over here. And that's actually a general idea, and I'll do it on the side here just to  prove it to you. If I have, let's say that these 2 angles-- let's say that the measure of that angle is

a, the measure of that angle is b, the measure of this angle we know is going to be 180 minus a  minus b. That's this angle right over here. And then this angle, which is considered to be an  exterior angle. So in this example, y is an exterior angle. In this example, that is our exterior  angle. That is going to be supplementary to 180 minus a minus b. So this angle plus 180 minus a minus b is going to be equal to 180. So if you call this angle y, you would have y plus 180 minus  a minus b is equal to 180. You could subtract 180 from both sides. You could add a plus b to both sides. So plus a plus b. Running out of space on the right hand side. And then you're left with--  these cancel out. On the left hand side, you're left with y. On the right hand side is equal to a  plus b. So this is just a general property. You can just reason it through yourself just with the  sum of the measures of the angles inside of a triangle add up to 180 degrees, and then you  have a supplementary angles right over here. Or you could just say, look, if I have the exterior  angles right over here, it's equal to the sum of the remote interior angles. That's just a little  terminology you could see there. So y is equal to a plus b. 114 degrees, we've already shown to  ourselves, is equal to 64 plus 50 degrees. But anyway, regardless of how we do it, if we just  reason it out step by step or if we just knew this property from the get go, if we know that y is  equal to 114 degrees-- and I like to reason it out every time just to make sure I'm not jumping to conclusions. So if y is 114 degrees, now we know this angle. We were given this angle in the  beginning. Now we just have to figure out this third angle in this triangle. So if we call this z, if  we call this question mark is equal to z, we know that z plus 114 plus 31 is equal to 180 degrees.  The sums of the measures of the angle inside of a triangle add up to 180 degrees. That's the  only property we're using in this step. So we get z plus, what is this, 145 is equal to 180. Did I do  that right? We have a 15, then a 30. Yep, 145 is equal to 180. Subtract 145 from both sides of this equation, and we are left with z is equal to 80 minus 45 is equal to 35. So z is equal to 35  degrees, and we are done.