CA Geometry: Exterior angles

We're on 41. Lea made two candles in the shape of right rectangular prisms. So I'm assuming when they say right rectangular prisms, they mean a kind of three dimensional rectangular shape. The first candle is 15 centimeters height, 8 centimeters long and 8 centimeters wide. So let's see, it's 15 centimeters high. So that's 15. 8 centimeters long. So maybe that's 8. And 8 centimeters wide. So maybe it goes back 8. So it looks something like that. That's candle number one. The second candle is 5 centimeters higher, but the same length and width. So the second candle is just 5 centimeters higher. So it looks something like this. Where this is still 8 and 8. But the height is 5 more than 15, so it's 20. Fair enough. How much additional wax was needed to make the taller candle? So if you think about it, we just have to think about how much incremental volume did we create by making that section five centimeters higher? So this candle, you can kind of view it as going up to here. It's 15 centimeters high. And then we added 5 right here. So what's the volume of this volume right there? So it's 8 by 8 by 5. So 5 times 8 times 8. 5 times 8 is 40 times 8 is 320. So we have to add 320 cubic centimeters more of wax to make the taller candle. Problem 42. Two angles of a triangle have measures of 55 and 65. Which of the following could not be a measure of an exterior angle of the triangle? So I think this is a good time to introduce what an exterior angle even is. So if I draw any polygon, and I'll draw a triangle since that's what this question is about. So let's say that that's my triangle. An exterior angle of one of the vertices is, you essentially extend one of the lines of the vertices out. So this is an interior angle right here. The exterior angle is if you extend this line out, so if I were to draw a dotted line that extends out this bottom line. This is the exterior angle right here. As you can see, it's going to be the supplement to this interior angle. And we could have extended the line out there. Or we could have extended this line this way. And we could have used this one. But we wouldn't add these two if we wanted to find all of the exterior angles. The exterior angle of this vertex right here is either this one or this one. And they are the same because both of these are supplements of this angle. This angle plus either of this angle or that one will add up to 180 degrees. So that's what an exterior angle is. So let's go back to the question. Two angles of a triangle have measures of 55 and 65. So let's say this is 55 and this is 65. Which of the following could not be a measure of an exterior angle of the triangle? Well we can figure out all of the exterior angles. So first of all what's this third interior angle going to be? Well they all have to add up to 180. So let's call that x. So we know that x plus 65 plus 55 is equal to 180. 65 plus 55 is 120. 120 is equal to 180. So x is equal to 60 degrees. So this angle right here, I'll do it in another color, this is 60 degrees. So what are all the possible exterior angles. So if I extended this line out like I did in the example of when I defined what an exterior angle is, this exterior angle would be 120 degrees. If were to do it here, if I would extend this out right here, what would this exterior angle be? Let's see, this plus 65 is 180. What's 180 minus 65. 180 minus 60 is 120, so this would have to be 115. So that exterior angle is 115. And then this one, let's see if I extend it out. One of the two lines that form the vertex. This is going to be supplementary to 55. So 180 minus 55 is 125. 180 minus 60 would be 120, and then it's only 55 so 125. So the three supplementary, or the three exterior angles of this triangle are 125. And they want to know what could not be a measure. So 125 is a measure of an exterior angle. So is 115. And so is 120. So our answer is D. None of the exterior angles are equal to 130 degrees. Problem 43. OK, they say the sum of the interior angles of a polygon is the same as the sum of its exterior angles, what type of polygon is it? And this is an interesting question. And it's something to experiment with for yourself. But I want you to draw random polygons with angle measures, because you know what the angles all have to add up to in a polyogon. And I think you'll find, that no matter what polygon you draw, all of the exterior angles are going to add up to 360 degrees. In fact, in that example we just did, what were they they were, for that triangle. If I remember, it's 115, 125, and 120. This was for a triangle. If you added them up, you get 5 plus 5, 10. And then that's 6. 360 degrees. For that triangle, which had kind of strange angles. It wasn't like an equilateral triangle or anything beautiful. And it's also the same if I were to draw a rectangle. Well let me not draw a solid rectangle. So if I have a rectangle like that. What are the exterior angles here? Well, I can continue this line right here. This angle right here is going to be 90. I could go either way, I could continue this up, but you can only do it once though for each of the vertices. Well that exterior angle is 90. I could go like that, that exterior angle is 90. I could go like that. That exterior angle is 90. So once again, 90 plus 90 plus 90 plus 90 that's 360 degrees. So it's a good thing to know that the sum of the exterior angles of any polygon is actually 360 degrees. And maybe we'll prove that in another video for a polygon with n sides. But now that we know that, so they say that the sum of the interior angles of a polygon is the same as the sum of its exterior angles, this is the same as saying the sum of the interior angles is equal to 360. Because this is always going to be 360 degrees no matter what the polygon is. So they're essentially saying what polygon's interior angles add up to 360 degrees. And that of course is a quadrilateral. My mouth got ahead of me. And if you think in a quadrilateral you have 90, 90, 90, 90 and they add up to 360 degrees. Next question. Let me copy and paste a couple of them so I don't have to keep doing this. OK. All right. What is a measure of angle x. So this is an exterior angle to the vertex B. So how do we figure this out? Well there's kind of a fast way and a slow way. And the slow way is to figure out this angle. Because you know that the sum of the angles add up to 180. And you say oh, x is going to be 180 minus that. Let's just do it the slow way and I think you'll see the intuition of a slightly faster way you could have done it. This plus 60 plus 25 is 85 degrees. Let's call this angle y. So we know that y plus 85 degrees is equal to 180. I got this 85 just by adding 60 to 25. So this is just saying that the interior angles of a triangle add up to 180 degrees. And we could figure out y right now. You could subtract 85 from both sides and you'd get y is equal to 95. And then we could figure out x from y, because x is the supplement of y. So then you could say x is equal to 180 minus 95 and you'd get 85. And that would be fine, it didn't take you too long, C is the answer. But a slightly faster way of saying it. OK, y plus 85 is equal to 180. And you also know that y plus x is equal to 180. So clearly, x is equal to 85. If you add 85 to y, you get 180, if you add x to y you get 180. So x would be 85. That would be a slightly faster way of thinking about it. But either way is fine if you're not under time pressure. OK, problem 45. If the measure of an exterior angle of a regular polygon. Regular polygon, so that means that all of the angles are congruent. Of a regular polygon is 120 degrees, how many sides does the polygon have? So this is the vertex in question, let's say that's the vertex of this polygon we're thinking about. We want to measure its exterior angles. So I'd extend one side of the vertex. And they're saying that that is 120 degrees. That tells me that the interior angle at that vertex is 60 degrees. It's the supplement to the exterior angle. So what regular polygon has all of its sides equal to 60 degrees? Well, the equilateral triangle. Regular polygon, all the angles and all the sides are congruent. So an equilateral triangle looking something like that would do the trick. It's a regular polygon, all the sides are the same. And its angles are 60, 60 and 60. So when they say how many sides does the polygon have? It has three, it's a triangle. I'm out of time. I'll see you in the next video.