All right, we’re in problem 17. It says which of the following best describes the triangles shown below? Okay, they want to know are they similar, are they congruent, etcetera, etcetera. Okay. So, let’s think about it a little bit. It tells us that this is 60-degree angle. This is a 90-degree angle all right they do this little square thing that tells us 90-degree angle. The angles on a triangle had to add up to 180.
So 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 or by the same argument this is going 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 you know if you just were to eyeball it, if you said okay the side opposite the 90-degree these are the corresponding sides, the ratios are equal but we see that they give us the actual length, then the hypotenuse of both to this triangles is 8, so the ratio is actually one to one. And when the ratio of the sides is one to one with the sides are actually congruent and if you’ve just given one side that’s enough then you could actually figure out the rest of them using—well you could use 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 two of the corresponding sides or 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, which of the following statements must be true if triangle GHI is similar so when they just write a curly thing like that without 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, you know before looking at the choices that means that the ratio of all of 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 schilling, now you can have similar triangles that are isosceles or equilateral, that’s not right. The two triangles must have exactly one acute angle. The two triangles must have exactly one acute angle. No, they can have two acute angles. They could have three acute angles. I mean the way that they have drawn here actually all of them are acute, there’s 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 are hard to process. Anyway, see at least one of the sides of the two triangles must be parallel. I don’t care how they are oriented, at least one of the sides of the two triangles must—I don’t know. You don’t care about the orientation of the triangles. The corresponding sides of the two 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 the—so this is almost you know the definition of a similar triangle.
Question 19, in the figure below AC is congruent to DF. Okay, so they are 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’re just going to give us one side and one angle. If they gave us another side, if they said that DE is congruent to AB that would be pretty cool. If they gave us this angle, if they said angle F is congruent to angle C that would be good. Let’s see what they gave 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 “Oh I have a side, an angle and the sides”, so you’d say by SAS, by side angle side. I know that these two triangles are congruent. So, AB is congruent to DE. Let’s see the other the one just so we didn’t miss anything.
AB is congruent to BC. 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. We see this is another time that I have a slight problem with the way they are going with this because if BC were congruent to EF, if this were true BC were congruent to EF, let me think about that. Can I draw this triangle in a way where they still not congruent. Because I have this angle here constraining it right. They told us that so its not like I can draw this line, this FE line, it’s not like I can draw it coming out here right because if I came out here then DE would have to come like that and then this angle couldn’t be what it said that they said it was.
So, I’m just trying to think. I should think that would be sufficient if you’re given that this side is congruent to that side. And then you can 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 again. Let’s see, those like a choice D. BC is congruent to DE. Well, these aren’t different corresponding sides so that’s clearly useless. I have a suspicion that this would have also been enough to prove but anyway I’ve all ready—I’m slightly, you know, 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 where you can spout out you know side angle side, angle side angle and things like that. And you’re going to forget those about three hours after you take the test and that’s pretty useless.
What useful is if you know something it gives you intuition about triangles that’s going to be useful for you on the SAT, that’s going to be useful before you take the essay, 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 you know memorize this theorems and all of that and even some of this notations never shows up again in your mathematical careers even if you do a piece G 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 the stuff at minimum just to jump through the hoop 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 alright problem 20, given AB and CD intersect at point E all right. AB and CD intersect at point E and just another side. I think you can even tell from my tone that I enjoyed the SAT problems a lot more because in some ways and in fact in everyway, the SAT problems really test your understanding of geometry but never do they mention the words similar, congruent, SAS, ASA, they never mention all of these things if 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 in the SAT, I know people who could do the other way and frankly I’d rather hire the person who does want the SAT team because that’s the person who I think has the intuition but anyway we have to do this and I probably shouldn’t rent like that.
See, given AB and CD intersect at point E. Angle one, okay given that AB and CD intersect at point A, fair enough and it tells us that angle one is congruent to angle 2. So, angle 1 so that and that are equal, all right so all ready that 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 right. That those two are parallel because if you view DC is a transversal, that you see that that’s a transversal between these two lines and because your alternate interior angles are the same or they are congruent, you know that those are going to be parallel lines but anyway I don’t know if that’s all useful. Were they going to ask us?
Which theorem or postulate can be used to prove that AED is similar to BEC? Okay so let’s see. Similar so I didn’t have to even say that those are parallel lines. So, if we know that—so what did they tell us? They intersect to point A. Okay so first one 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 then you know if you know two of the angles in a triangle you know the 3rd so this angle and that angle have to be the same. But in general, if you know that two angles of a triangle is the same, the third has to be the same. So, that tells you that’s a similar triangle. So, we could use angle angle. We know that two angles are same as two other angles so we know we’re dealing with similar triangles. Anyway, all out of time because of my rent, see you in the next video.