 Okay, we've talked about right angle triangles and how to solve right angle triangles. Now with triangles, you don't always get right angle triangles. So you're gonna have to figure out how to solve certain triangles that aren't right angles. What we're gonna do is talk about congruent triangles and similar triangles. So we'll talk about congruent triangles first, and then we'll talk about the similar triangles, okay? Congruent triangles, when they say, or when they ask you a question, if two triangles are congruent, what they're asking you is, are two triangles identical? That means are they equal? So for example, they're gonna give you a triangle and say, okay, you have this triangle and is it congruent to this triangle? Now, this is the congruency side, which basically means if this triangle is equal to this triangle. Now, the only way for you to know if two triangles are identical is if they have the same three sides, which is called side, side, side, if they have side, angle, side, or if they have angle, side, angle. Now keep in mind, angle, side, angle is also, it also occurs when you have angle, angle, side. Now, we'll talk about this, but basically you should remember this because if you're given two angles in any triangle, you know the third angle. So that would mean this isn't just angle, angle, side, it's actually angle, side, angle. So for example, if they give you two triangles and they give you the following, following information, you're gonna have to be able to tell if these two triangles are identical or not. So if they tell you if this angle is equal to this angle and this angle is equal to this angle, they're gonna ask your question is, are these two triangles congruent? That means are they identical? Well, according to what we know, there's only three situations where you can have two triangles equal to each other. Side, side, side, side, angle, side, angle, side, angle. Now how many pieces of info have they given you here? They told you two angles here, two angles here. If they've told you that, then they've also given you the third angle because we know the sum of the angles in a triangle is 180. So if you go 180 minus these two, you're gonna get this angle, 180 minus these two, you're gonna get this angle. So if these two are the same as these two, then this angle must be the same as this angle. Now this, these two triangles are similar. They're not congruent. We'll deal with the similar triangles a little bit later on. We're doing ratios trying to figure out what each side is. So one thing to keep in mind is for the only way for you to find out if two triangles are congruent is you need to know at least one thing about a side. There's got to be a side in there, okay? You don't necessarily have to have any angles because this one doesn't have any angles, but you need to know at least one side. So if they say this angle is equal to this angle, this side is equal to this side, and that angle equals that angle, then these two triangles are identical. They're congruent, okay? The reason being is you have angle, side, angle, angle, side, angle, which is angle, side, angle. Now in this situation, if they give you, let's say this angle equals this angle, right now you don't have angle, side, angle because they have to follow each other. You can't skip anything. So right now you have angle, angle, side, and you're skipping the side. So you have neither of these three situations. But the sum of the angles in a triangle equals 180. So if you know these two angles, you know these two angles, then you know this angle because you know these two angles because the sum of the angles in a triangle is 180. So you would know that this, what this angle is, and because these two are the same as this, then you know that this is identical to this. Now you have angle, side, angle, angle, side, angle. So that's why angle, angle, side is the same thing as angle, side, angle, okay? Now let's deal with some of the other guys. Here's our triangles. Now if they tell you this side is equal to this side, this side is equal to this side and this side is equal to this side, then these two triangles are congruent because you have side, side, side. Their sides are going to be same. So if they tell you that this is five, this is six and this is seven, then you know that this is five, this is six and this is seven. Now this may look like a 90-degree angle, but I haven't marked it as 90 degrees, so you can't say that this is, ah, the hypotenuse should be larger. If I drew a 90-degree angle there, then this triangle would not make sense because this would have to be the longest side. We talked about this in the right triangle, but keep in mind if they tell you this and they tell you that these sides are the same as these sides, then you know what these sides are. They can also give you, they could say, here's two triangles, they could say this side is equal to this side, this side is equal to this side and this angle is equal to this angle. Well what you have here is side angle side, side angle side, side angle side. So these two triangles would be congruent. Now if they gave you not this angle, but this angle, you wouldn't have any of these situations because this is angle side, you're missing an angle, then you have another side. You don't have any of those situations. They have to follow exactly the way it's written. So if they gave you this situation, these two triangles are not congruent. Well they could be congruent, you don't know. So if you don't know, as far as you're concerned, they're not congruent. So basically it's the simplest way we know how and the only way we know how to identify two triangles as being equal, which is one of these three situations. Now let's talk about congruent triangles, let's talk about similar triangles. Now with similar triangles, what you must have is angle, angle, angle. This would mean that the two triangles are replicas of each other, but one is bigger than the other one. So for example if you had this triangle and if you had this triangle and if they said this angle equals this angle, this angle equals this angle, and that angle equals that angle, then you would say these two angles are similar. They have the same ratio, same proportions, but one is a smaller version of the other angle. So if they tell you that these are all angles are equal, and if they say this side is 5, this side is 6, and this is 10, and that's 8. So what they're going to do is say, hey, what are the other sides of these triangles? Now what happens here is because you don't know if any of these angles are 90 degrees, you can't use Sokotoa. Sokotoa if you remember only applies to 90 degree triangles. So right now the only way you can figure out what these triangles are is the relationship between the two sides that are across from the angle you know. So what you need to have is a connection. Something connecting this triangle to this triangle. You know that the angles are the same, that's fine. But you also have to know where the sides are related to each other. So this guy becomes, oh, this is across from this angle which has the triple lines on it. This side is across from this angle which has the triple lines on it. And if you remember for any triangle, we'll bring up the drumsticks again. For any triangle, the side that this angle controls is the opposite from it. So this angle is related to this side and this angle is related to this side. So once you know that, then you know that this side represents this side. So you can use ratios to solve this. What you can do is say, okay, 5 over 10 must be equal to the ratios of the other sides which is, that's across from the double tick angle. And that's across from the double tick angle. So 6 over x. So 5 over 10 must be equal to 6 over x. And for this one, it would be, hey, the single tick across from the single tick must be equal to w over 8. Now you can't solve this equation all through.