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CA Geometry: More on congruent and similar triangles

17-20, more similar and congruent triangles. Created by Sal Khan.

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Video transcript

All right. We're on problem 17. It says, which of the following best describes the triangles shown below? OK, they want to know, are the similar? Are they congruent? Et cetera. They tell us that this is a 60 degree angle. This is a 90 degree angle, they do this little square thing there. The angles in a triangle have to add up to 180. So if this is 90, this is 60, that adds up to 150. So this has to be 180 minus 150. So this has to be a 30 degree angle. So that has to be 30 degrees right there. Fair enough. Now let's get this one. That's 30, that's 90. Well, by the same argument, this has got to be 60. Because they all have to add up to 180. All right. So just there we know that all of the angles in both of the triangles are congruent. Or that their measures are equal. So we know already that these are definitely both similar triangles. Now, a similar triangle also tells us that the ratio of all of the sides are equal. So. if you just sort of eyeball it, if you said, OK, the side opposite the 90 degree, these are the corresponding sides, the ratios are equal. But we see that they give us the actual lengths. So the hypotenuse of both of these triangles is 8. So the ratio is actually 1:1. And when the ratio of the sides is 1:1, when the sides are actually congruent, and if you're just given one side, that's enough. Then you could actually figure out the rest of them using a little trigonometry or something like that. We're not going to go there just yet. But in geometry class, you learned that if something is similar and and at least one of the corresponding sides is congruent, then the whole thing is going to be congruent. So these are both similar and congruent triangles. Both similar and congruent, that's A. Problem 18. OK, let me cut and paste it. Which of the following statements must be true if triangle GHI is similar. So if they write this that means congruent. If they just write that, that means similar. Which of the following statements must be true if triangle GHI is similar to triangle JKL? So even before looking at the choices, that means that the ratio of all the sides are the same, or all of the angles are the same. Let's see what they give us. The two triangles must be scalene. Now, you have similar triangles that are isosceles or equilateral. That's not right. The two triangles must have exactly one acute angle. No, they could have two acute angles. They could have three acute angles. The way that they've drawn it here, actually all of them are acute. None of these angles are greater than 90 degrees just the way they've drawn. So that's not right. Some of these statements are so crazy that they're hard to process. Anyway, C, at least one of the sides of the two triangles must be parallel. I don't care how they're oriented. You don't care about the orientation of the triangles. The corresponding sides of the triangles must be proportional. Yeah, that's one of the ways that you know that something is similar. That the corresponding sides are proportional. So that is, D. So this is almost, do you know the definition of a similar triangle? Question 19. Let me erase this. OK. I have copied it. Now I am pasting it. In the figure below, AC is congruent to DF. OK, so they're equal to each other. AC and DF are congruent. And angle A is congruent to angle D. Fair enough. That's angle A, that's angle D. That's what they tell us. Which additional information would be enough to prove that triangle ABC is congruent to DEF? So they just gave us one side and one angle. If they gave us another side, if they said that DE is congruent to AB, that'd be pretty cool. If they gave us this angle, if they said angle F is congruent to angle C, that'd be good. Let's see what they give us. AB is congruent to DE. Yeah, sure. If AB is congruent to DE then we definitely have congruent triangles. And you know the theorem that you would have to say in your geometry class is, I have a side, an angle, and a side. So you would say by SAS, by side, angle, side, I know that these two triangles are congruent. So AB is congruent to DE. Let's look at the other ones to make sure we didn't miss anything. AB is congruent to BC. Well, that's fine. But that doesn't tell us how AB relates to DE. So that's a useless statement. BC is congruent to EF. Well, see, this is another time that I have a slight problem with the way they're going with this. Because if BC were congruent to EF. Let me think about that. Could I draw this triangle in a way where they're still not congruent. Because I have this angle here constraining it. They told us that. So it's not like I can draw this line, FE, coming out here. Because if it came out here, then DE would have to come like that. And then this angle couldn't be what they said it was. So I'm just trying to think, I actually think that would be sufficient. If you're given that this side is congruent to that side. I think you could make a trigonometric argument very easily to show that these two triangles have equal sides. But anyway, I'm not going to bother with that. Let's see. Let's look at choice D. BC is congruent to DE. Well, these aren't even corresponding sides. So that's clearly useless. I have a suspicion that this would have also been enough to prove. But anyway, I don't want to insult anyone in the California Department of Education, but I'm slightly disappointed by some of these questions. Because I feel like they really aren't testing intuition, they're just testing to see whether you know the definitions of some of these geometric terms. And whether you can spout out, side, angle, side, and angle, side, angle. And things like that. And you're going to forget those about three hours after you take the test. That's pretty useless. What's useful is if you know something that gives you an intuition about triangles. That's going to be useful for you on the SAT, that's going to be useful for you when you take trigonometry. And I'll tell you a dirty secret. You will never use ASA theorem or SAS theorem or anything like that again in your mathematical careers. Your 9th or 10th grade geometry class is the first and the last time that you'll ever see them. So I have a slight problem where they want you to memorize these theorems and all of that. And even some of this notation never shows up again in your mathematical careers. Even if you do a PhD in mathematics. The only time you'll probably see it again is if you become a geometry math teacher. Anyway, but it's good. I mean you should know how to do this stuff at minimum just to jump through that loop that society makes us all jump through. So problem 20. You don't want someone else to get paid more just because they were willing to say SAS, ASA. Anyway, all right, problem 20. Given AB and CD intersect at point E. And just another aside, I think you can even tell from my tone that I enjoy the SAT problems a lot more. Because in some ways, in fact, in every way, the SAT problems really test your understanding of geometry, but never do they mention the words similar, congruent, SAS, ASA. They never mention all these things that you essentially memorize in your geometry class. And I know tons of people who get A's in geometry and then they don't do well on the SAT. And I know people who do the other way. And frankly, I'd rather hire the person who does well on the SAT. Because that's the person who I think has the intuition. But anyway, we have to do this. And I probably shouldn't rant like that. Given AB and CD intersect at point E. Fair enough. And they tell us that angle 1 is congruent to angle 2. So that and that are equal. All right, so already those look like alternate interior angles. If this line were parallel. In fact, I think that's enough to show that this line is parallel to this line. Because, if you view this as a transversal, if you view DC as a transversal, then you see that's a transversal between these two lines. And because the alternate interior angles are the same, or they're congruent, you know that those are going to be parallel lines. But anyway, I don't know if that's at all useful. What are they going to ask us? Which theorem or postulate can be used to prove that AED is similar to BEC. OK. So let's see, so I didn't have to even say those are parallel lines. So what do they tell us? First of all, we know that 3 and 4 are congruent angles, because they're opposite angles. Once again, I don't like the word vertical angles, because these angles are clearly not vertical. They're more side by side. But they're definitely opposite. So those two angles are the same. 1 and 2 are the same, 3 and 4 are the same. If you know two of the angles in a triangle, you know the third. So this angle and that angle have to be the same. But in general, if you know that two angles of a triangle are the same, the third has to be the same. So, that tells you that it's a similar triangle. So we could use angle, angle. We know that two angles are the same as two other angles. So we know we're dealing with similar triangles. Anyway, all out of time because of my rant. See you in the next video.