Were on problem 21, in the figure below n is a whole number all right. What is the smallest possible value for n? Okay and both of these sides are n and so this is something it’s actually really good to get this intuition because this shows up on all such a standardized test. And so let’s think about how small can we make it? So, the lower this kind of the top of this pyramid becomes a smaller n become right if we push this top of the pyramid really high then n would have to be really big right. So if we take it, for example if we made the triangle into that then clearly this length is shorter than that link and we want to keep glorying it to get a small possible n but what happen if we lower it all the way, if we like to flatten this triangle right, if we just flatten it all take down.
So essentially, this would be the top of it and this would be n and this would be n. I hope you’re be visualizing that properly, they’re flatten at the triangle. So, these two sides would just go flat with the base, right. And so this top if I will do it in magenta, it’s right here. So, this is as small as n could get out of you know one can argue whether this is a triangle at all anymore. It’s really aligned now because I’ve squished out all of that area in there but even in this case, n would have to be the smallest case so it would be 7.5, each n would be half of this 15.
So, that’s really you know as we make this, as we push this base down, that’s kind of the limit that n approaches. N cannot be any smaller than 7.5 and they tell us that n is a whole number, right. So, n has to be greater than 7.5 in order for this to be a triangle right and n is a whole number so n is equal to 8 and that’s choice C and that should import into which we have in general that the third side of a triangle can never be bigger than two of the other sides combined then you’re dealing with something else. You’re not dealing with a triangle. Even at the third side is equal to the other two sides then you’re actually dealing with the line because you’d have to squish out all of the area of the triangle in order to get there. Anyway then, I like that problem.
Next question, I think just eyeballing it, they want us to do the same thing, same type of intuition. And I have my rent in the last video about how they weren’t doing problems with that that give you intuition or the tester intuition but now I’ll take that back because I think that’s what they are testing now. Which of the following sets of numbers could represent the lengths of the sides of a triangle? Two and two and five, so this is the same thing again. How can I have two sides of a triangle combined being sure of the third side if I decide of length two and then I had another side of length two, there’s no way that this last side can be five, right. Even if I completely flatten this triangle two and two the longest at this last side this third side could be is four so that can be a triangle. Same here, let’s look at the other one. I mean 3, 3, 5, that’s no reason why that can’t be a triangle, 3 and 3. Right 3 and 3 is six so you know if I flatten that out a lot I mean then the side can be as long as 6 and obviously I can squeeze them together like this, 3 and 3, and then this little third side could be something really small because if you see anything in between you know zero and six so obviously it can be five so that’s going to be the answer. 4, 4, 8 same problem, I would have to squish the triangle so much if those are two of the sides 4 and 4. This last side is still going to be less than 8. The only way we get to 8 is if I push the top of this triangle all the way down in such to make a line.
Once again, one side of a triangle can't be bigger than two of the other sides combined. It actually can't be equal to two of the other sides combined then we’re dealing with a line, so they really testing that same intuition and the choice is B.
In the accompanying diagram parallel lines L and M are cut by transversal T since I get classic parallel line with the transversal problem and they’re parallel that’s why I did those arrows. Which statements about angle one and two must be true? All right so we could play, I don’t know if you’ve seen the conic academy videos on the angle game but that’s what we’re going to play here, the angle game. So, angle one if you want to look at its corresponding angle, its corresponding angle on the other parallel line or with the transversal and the other parallel, n is write there and they’re going to be congruent. Those two angles are going to be congruent so you can say this is equal to the measure of angle one. I’ve picked up their terminology well I think.
So, this is equal to a measure of angle one and this is obviously angle two. You see immediately that there have to be supplementary right because when you add them together you get a 180 degrees, alright. Together they go all the way around and they kind of form a line. So, you know that if this angle and this angle are supplementary and this angle is congruent to angle one then angle one and angle two must be supplementary. So, what did they say? So, angle one is definitely not necessarily congruent to angle two, right it’s congruent to this angle here. Angle one is a compliment of angle two. Compliment means you add up to 90, no, we’re talking about supplement, it’s not that. Alright, angle one is a supplement of angle two, there you go and there’s nothing that says at the right angle, that’s silly.
All right, next problem, what values of A and B make the quadrilateral M N O P a parallelogram. Okay, for this to be a parallelogram, the opposite sides have to be equal and I challenge you to experiment to draw a parallelogram where opposite sides are parallel, where the opposite sides are also not equal. If you make two of the sides not equal then the other two lines are going to be a parallel anymore and you can play with that if you like but if opposite sides are going to be equal, that means 4a + b = 21 right because they are opposite sides so they should be equal to each other. Similarly, 3a - 2b = 13 because they are opposite sides. So, 3a - 2b = 13 and now we have two linear equations with two unknowns so this is really an Algebra 1 problem in disguise so let’s see. If we want to—and they want to solve for both, let’s see if we can cancel our b. So, let’s multiply this top equation by two. So, times two and I’m doing that to cancel out the b’s so you get 8a + 2b = 42 and I did that so that two b and the minus 2b cancel out. So, let’s add these two equations to each other, the left hand side 3a + 8a = 11a, the b’s cancel out -2b + 2b that’s no b’s = 42 + 13 = 55. Look that well divide both sides by 11, you get a = 5. Now, if a = 5 what’s b? Let’s substitute back at that first equation because I’ll pick either. So, 4 x 5 + b = 21, 20 + b = 21 so subtract 20 from both sides b = 1. So, a = 5, b =1 that is choice b.
Problem 25, quadrilateral A B C D is a parallelogram. If adjacent angles are congruent, which statement must be true? All right, so let me draw a parallelogram. So, parallelogram says that all of the sides are parallel, opposite sides are parallel, that’s parallel to that and that’s parallel to that all right but they gave us another statement. Let’s say if the adjacent sides are congruent, so they’re saying that this is congruent to that. They say if adjacent angles are congruent so I don’t know if they are saying it’s just one of them or all of them or all of them but it’s the same thing actually. That angle is congruent to that angle something interesting has to happen. They both have to be 90 degrees and I want you think about that a little bit. So, let me just draw the kind of bottom part of that parallelogram. So, this is—and I drew it more so if you say that this line right here is a transversal and that since that’s parallelogram we know that this line is parallel to that line. So, that line is parallel to that line.
So, we have a transversal between parallel lines, this angle is congruent to that angle. They are corresponding angles and this angle is the supplement to this angle right. They have to add up to 180 so this angle, this red angle of this brown angle have to be supplement if we said angle one and angle two. They have to add up to 180. Measure of angle one plus measure of angle two have to be equal to 180 but then they tells us even more. They tell you that they are congruent. These are adjacent angles right, that’s angle one and that’s angle two, they’re adjacent. So, if both of these are congruent, if their measure is equal to each other they both have to be equal to 90.
So, if adjacent angles are congruent to the parallelogram then these angles are 90 and these angles are 90, those are going to be 90 by the same argument. So now, we’re not dealing with just a parallelogram. It’s a rectangle, this is quadrilateral A B C D square, they could be a square but it didn’t say—in order to be square it have to tell us that all the sides are equal to each other. So, it’s not A, I mean that’s definitely, that could happen but don’t have to be. A B C D is a rhombus and rhombus all the sides have to be equal to each other, they didn’t tell us that. A B C D is a rectangle, sure because we know all the angles now are 90 degrees. And that is choice C. See you in the next video.