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Missing angles (CA geometry)

46-50, deducing the measure of angles. Created by Sal Khan.

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Video transcript

We're on problem 46. In the figure below, line segment AB is parallel to C. I think there's a typo here. I've got to believe this has got to be CD. That's got to be a typo. All right, so AB, I'm assuming, is parallel to CD. That's probably what they wanted to have, because you can't be parallel to a vertex. And they say, what is the value of x? So the way to think about it is, if these two are parallel lines, then this segment up here is a transversal. If we just extend the two lines like this, let me see if I can do it well. So if I extended that line like that, and if I extended this line like that, then segment AD is just a transversal. Let me see if I can do that one right. So if I just extended it like that, you see it's a transversal, and I could go the other direction as well. I think you get the idea. So if that is a transversal, what do you know about transversals? We know that this angle right here, I'll draw it small, is congruent to this angle over here. So the measure of this angle is also x plus 40. Because they're corresponding angles, and you could see that by inspection, and if you moved around the transversal, it would make sense that that's the case. So this is x plus 40 and this is minus 40, and they're clearly supplements of each other. They're supplementary angles. Then the sum of these two angles have to be equal to 180. So let's figure it out. So x minus 40 plus x plus 40 is equal to 180, because they're supplementary. The 40's cancel out. So you just minus 40, plus 40 adds up to zero. So you're left with 2x is equal to 180. x is equal to 90. So it's D. 47: The measures of the interior angles of a pentagon are 2x, 6x, 4x minus 6, 2x minus 16, and 6x plus 2. What is the measure in degrees of the largest angle? OK, so first of all, we have to remember what is the sum of the interior angles of a pentagon? And that's where I always draw an arbitrary pentagon. Let me see if I can do that. Actually, there's a polygon tool here. How does it work? I'm just trying to draw a pentagon. I don't know if that's any different than the line tool, but regardless. So how many triangles can I draw in a pentagon? And that tells me what my total interior angles are. And there is a formula for that, but I like relying on your reasoning more than the formula, because you might forget the formula, or even worse, you might remember it, but not have the confidence to use it, or you might remember it wrong ten years in the future. So the best thing to do, if you have a polygon, is to count the triangles in it. Straightforward enough. It's almost easier than using the formula. So a pentagon has three triangles in it. So the sum of its interior angles are going to be 3 times 180, because it has 3 triangles in it. Each triangle has 180 degrees. And I know you can't see what I just wrote. So the sum of all of these angles are going to be the sum of all of the angles in all three triangles. So it's 3 times 180 is equal to 540 degrees. So that's the sum of all of these interior angles. Now, they say that each of them are 2x, 6x, et cetera. So the sum of all of these terms have to be equal 540. I'm going to write them vertically. It makes them easier to add. So if we write there 2x, 6x, 4x minus 6, 2x minus 16, and 6x plus 2. This is going to be the largest, right? That sum is going to equal 540. So let's add this up. Minus 6, minus 16, that's minus 22. Plus 2 is minus 20. That's right. And 2x plus 6x that is 8x plus 4 is 12, 12 plus 2 is 14, 14 plus 6 is 20. So we have 20x minus 20 is equal to 540 degrees. Let me write it again. 20x minus 20 is equal to 540. Let's divide both sides of this equation by 20. So you get x 1 minus 1 is equal to-- it would be 54 divided by 2, which is equal to 27. Add 1 to both sides, x is equal to 28. And they want to know, what is a measure in degrees of the largest angle? That's going to be this one. That's the largest one. It's 6 times x plus 2. So 6 times 28, that's 48. 2 times 6 is 12 plus 4 is 168. So it's 168 plus 2. It's 170 degrees. Choice C. Problem 48: What is the measure of angle 1? So this we're going into the angle game. And these are fun, because they are kind of these deductive reasoning problems where you just use a couple of simple rules and just fill in the whole thing. So let's think about it. This is 36 degrees. They tell us that this whole angle right here is a right angle. So this angle right here is going to be the complement to 36 degrees. 36 plus this angle have to be equal to 90. So what's this one? This one is 90 minus 36, which is 54. That's going to be 54 degrees. 90 minus 30 is 60. Right, that's 54. And this angle right here, that's going to be the supplement of 88. So this is going to be-- I'll do it in a different color-- 180 minus 88. That is equal to 92 degrees. Now this angle 1 plus the 54 plus the 92 is equal to 180. So we know that-- let's say angle one plus 54 plus 92 is equal to 180. This is 146 is equal to 180. Subtract 146 from both sides. The measure of angle one is equal to 80 minus 40 is 40, so 80 minus 46 is equal to 34 degrees. So the answer is A. Problem 49: What is the measure of angle WZX? So they want to know what this angle right here is. Let's do the angle game some more. Let's see, we can immediately figure out what this angle is, because it is the supplement of 132 degrees, so this is going to be 180 minus 132. So this is 48 degrees. This angle plus this angle is going to be equal to this angle. Or this angle plus this angle plus this angle is equal to 180. I don't know what I just said, I think I said something wrong. Write that down. So this angle is going to be equal to 180 minus 52 minus 48. Because the sum of the angles add up to 180, and so that is equal to 180 minus 100 which equals 80 degrees. So this angle right here is equal to 80 degrees. And the angle they want us to figure out is the opposite of this angle, or in the U.S., I guess, they say vertical angles. And so opposite or vertical angles are equal or they're congruent, so this is going to be 80 degrees as well. And that is choice A. Problem 50: What is the measure of an exterior angle of a regular hexagon? A regular hexagon tells us that all of the sides are the same, it's equilateral, and all of the angles are the same, equiangular. So if we just knew what's the total degree measure of the interior angles, we could just divide that by 6, and then that would give us what each of the interior angles are, and then we could use that information to figure out the exterior angles. Let's just do it. So once again, I like to just draw a hexagon. Let's just draw a hexagon and count the triangles in it. Two sides, three sides, four sides, five sides and six sides. And how many triangles do I have here? One, two, three. So I have one, two, three, four triangles. The sum of the interior angles of this hexagon, of any hexagon, whether it's regular or not, are going to be 4 times 180 and that's 720 degrees. And it's a regular hexagon, so all the interior angles are going to be the same. And there's six of them. So each of them are going to be 720 divided by 6. Well, 6 goes into 72 twelve times. So each of the interior angles are going to be 120 degrees. And I didn't draw it that regular, but we can assume that all of these are each 120 degrees. Fair enough. Now, if all of those are each 120 degrees, what is the measure of an exterior angle? Well, we could just extend one of these sides out a little bit. We could say, OK, if this is 120 degrees, what is its supplement? Well, these have to add up to 180, so 180 minus 120 is 60 degrees. I could do it on any side. I could extend that line out there, and I'd say, oh, that's 60 degrees. So any of the exterior angles are 60 degrees. B. All right, do I have time for one more? I'll wait for the next one in the next video. See you soon.