Hi everyone, welcome to this lesson where today we're talking about similar triangles. So we're going to be looking at three different similarity theorems and we're also going to be looking at some algebraic ways that we can prove whether or not triangles are similar, as well as actually doing a couple of real geometric proofs. So let's get started.
The first similarity theorem is called AA similarity and that stands for angle angle. So basically what this is saying is that as long as you have two triangles with two pairs of congruent angles, that is enough to prove that the two triangles are similar to each other. You don't need the third angle because we know the third angle theorem. So if you already have two pairs of congruent angles, then the third angle is definitely going to be congruent.
And that is enough to prove that two triangles are similar. And remember, similar means they're the same shape, but they're not the same size. So you could have one small little triangle and then one big, bigger triangle, and they could have the same angle measures, but they're just different sizes.
Something I always think about with this one is an equilateral. That's the easiest thing to think about. So remember an equilateral triangle is a 60, 60, 60. Well, you could have a tiny little 60, 60, 60 triangle, right?
And then you could have a really big one. And those are definitely similar to each other. They're still equilateral.
They still all have the degrees of 60, 60, 60, but they're just different sizes. Side, side, side similarity. So this is again, another way we can prove that two triangles are congruent, I'm sorry, similar, excuse me, if all three pairs of sides are proportional to each other, all pairs of corresponding sides. So if I'm able to set up my ratios from one triangle to the next, and all three are the same scale factor, then that is enough to prove that the two triangles are similar to each other.
So if I had a 3, 4, 5 triangle, and then a 6, 8, 10, 3, 4, 5, and 6, 8, 10. If I set up those ratios, it's just really a one to two ratio, and those would be similar. And the last similarity theorem is SAS. So remember side angle side when we were proving triangles were congruent? That meant that you had to have two congruent sides and a congruent angle that was included.
So the angle in between the two sides. But for similarity, it's about the included angle is still congruent, but the sides are proportional to each other. and it has to be side angle side so you have to have two sets of sides that are proportional to each other and then the included angle has to be congruent to each other so the moment you have proportional sides and that included angle that is congruent then you are good and they are similar so we're going to look at a bunch of different diagrams for the different similarity theorems so in this first one there's nothing to solve for but it's basically just saying hey if i looked at this diagram I see I have one pair of angles marked congruent and I have a second pair of angles that have the same measure. So by AA similarity, I would be able to make this statement.
And again, your similarity statement is incredibly important. Every letter has to match up. So A has to coordinate with D, B has to coordinate with E, and C has to coordinate with F.
In this diagram, based on the fact that we just said that two triangles with the same angles are Similar to each other. Well, here's what I have I have this small little triangle with an angle measure and the angle measure and then I have the big triangle with the same Angle measure. So this little triangle H I J is similar to triangle G I K and So basically if I wanted to solve for X Setting up these proportions from a ratio from one side of one triangle to a side of the other triangle well, if I wanted to set up a proportion, I could say the right hand side of the small triangle x over the top of the triangle eight would be equal to the entire right hand side of my bigger triangle g ik, which would be x plus six.
So that would be this entire length over the top of the triangle 12. And so I can see you know, what is my proportions, how I would set this up. So notice I took a side of a triangle over a side, and I set it equal to the same side of the bigger triangle over the other side. This is now just using our cross products to solve, and we end up getting a nice answer of x equals 12. In this next problem, also if I asked you to solve for x proportion-wise, again this is enough information to prove that these two triangles are similar.
So if they are similar, we know that the sides are proportional. So I know that 3 corresponds with 5 because notice it's the side in between the two angles that are marked congruent. And then that would relate to x plus 1 over x minus 2. So setting up that proportion. So notice I went from triangle to triangle equals triangle to triangle. So side to side is equal to side to side.
Then using my cross products, I'm going to solve this proportion. And I end up getting negative 11 halves for x. Okay, so now in this first diagram, there's clearly nothing to solve.
But just asking ourselves, is this proportional? So remember, if you're like, how do I know which sides match up with each other? It's always the smallest side of one triangle should be in relation to the smallest side of the other triangle. So 6 would correspond with 12, 7 would correspond with 14, and 8 would correspond with 16, which of course all are one half. So by side, side, side similarity, I can make that similarity statement here.
if I was checking to see, you know, are these sides proportional to each other? Um, well, six would correspond with four, 7.5 would correspond with a five and nine would correspond with six. Those all definitely check out.
They all become three halves. And so if this is X, I know that the angle measure here is also going to be X. That's what we would solve for.
Another way to think about it is this is 40. If I wanted to solve for x, I know this is 80. This is 80 because m corresponds with o. Just rewrite that. And so no matter how you look at it, you would be able to figure out, well, 80 plus 40 is 120. 180 minus that 120 gives me 64x. And this last one, if I wanted to set up my proportions, so five would correspond with 10, 12 goes with 24. Again, the medium-sized side would go with the medium-sized side in that triangle. And then my longest length to the longest length.
And those are definitely similar. And they are right triangles, okay? Those are Pythagorean triples of 5, 12, 13. This is actually just a multiple of that. triple so this is actually just a nice little right triangle okay side angle side so making sure that you have two pairs of sides proportional and the included angle is congruent so six over fifteen is definitely equal to eight over twenty those been both simplified to two-fifths so by side angle side similarity those two triangles are similar here if i set up that same ratio The smallest side of one triangle goes with the smallest side of the other triangle, 4. So 5 over 4 set equal to 6 over 5. Those are definitely not going to be the same. 5 over 4 is 1.25.
6 over 5 is 1.2. And so this would tell me these are actually not similar. Now this last one, okay, so by vertical angles, I definitely have congruent angles.
And now proportional 5 goes with 15, 7 goes with 21. So if I was looking at this, 5 over 15 is definitely equal to 7 over 21. And so therefore, if I set up that same proportion and I say, okay, well, it looks like I have a three to one scale factor here from LMN to PON. Then if OP is 4 and I multiply it by 3, I would get that X is 12. Okay, a couple of proofs. So here it says given triangle XYZ and triangle ABC are right triangles.
It also says here that XY over AB is proportional to YZ over BC. So that's great information. We're already given in this proof, hey, there's two right angles.
And we know what we're going to say. Right angles are all congruent to each other. And we're already given a pair of proportional sides.
And we have to prove that these two triangles are similar to each other. So, of course, we copy everything in the given into our first statement. Then because we are told that they are right triangles, then we're going to be able to say that, okay, well, then that means that angle Y and angle B.
are right angles because the definition of a right triangle and if they're both right angles then all right angles are congruent to each other. And now because we've officially made the statement that these two angles are congruent to each other and we were given the fact that we have a pair of proportional sides, we can say they are similar by SAS similarity theorem. Okay, so now this proof here.
So we are given that ABCD is a trapezoid. So think about trapezoid properties, right? One pair of opposite sides is parallel. And the other pair of sides is not parallel to each other.
We're not told it's isosceles, but just some things to think about. So we are told that DE, I'm sorry, we are trying to prove that DE over EB is proportional to CE over EA. Okay, so we're trying to prove.
prove that these are proportional to each other. So this is interesting. Now think about what comes to mind when you have to prove that two values are proportional to each other. That would mean that we're trying to prove that the triangles are similar to each other first. Kind of like when we had to prove two sides of a triangle were congruent, we had to prove that the two triangles were congruent.
And then by CPCTC, we were able to say that the two sides were congruent or two angles. So let's see where we can get started here. And also noticing what these are connected to, of course, notice that we are talking about now, two triangles, this is what I want you to see is that look, the red and the blue form a triangle down here, right?
Those are two sides of a triangle. And these are two sides of a triangle. So it's like, okay, if we can get those to be congruent to each other, then we can, I'm sorry, similar to each other, then we would be able to prove it. So let's see where we go with this. So, okay.
AB is parallel to DC. And we know that because it's just definition of a trapezoid. Okay. So now, because those are parallel to each other, what ends up happening is these lines here.
what looked like diagonals actually become transversals. So now we can say, okay, well, this angle BDC is congruent to angle ABD. Oops, made that one a little too long. And then this angle here, BAC is congruent to angle DAC. So now look at this.
Because of those diagonals in this trapezoid, we're now able to say, hey, these two triangles here definitely have two pairs of congruent angles. And now think about that. If I prove that these two triangles have two pairs of congruent angles, then I can say that those two triangles are actually congruent to each other by the angle-angle similarity theorem. And then if these two triangles are similar to each other, well, then that must mean that the sides are...... proportional.
And so I would be able to say DE over EB is equal to CE over EA. And that's because corresponding sides of congruent triangles are proportional. I know there's a lot going on in this video. I hope it was helpful for you.
Thank you so much for watching. Bye.