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Current time:0:00Total duration:10:53

Video transcript

we're on 41 lay I made two candles in the shape of right rectangular prism so I'm something a wreck when they say right rectangular prism they meet a kind of a three-dimensional rectangular shape the first candle is 15 centimeters high eight centimeters long and eight centimeters wide so let's see it's 15 centimeters high so that's 15 eight centimeters long so maybe that's eight and eight centimeters wide so maybe it goes back eight right so it'll look something like that that's candle number one the second candle is five centimeters higher but the same length and width so the second candle is just five centimeters higher so it looks something like this where this is still eight and eight but now this is this the height is five more than 15 so it's twenty 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 so that's 15 and then we added five right here so what's the volume of this of this volume right there so it's eight by eight by five so five times we could say five times eight is so five times eight times eight five times eight is forty times eight is 320 so we had 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 have draw any polygon and I'll draw a triangle since that's what this questions about doesn't excuse me so let's say that that's my triangle an exterior angle of one of vertices is you essentially extend the one of the lines of that 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 drawn 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're the same because both of these are supplements of this angle right this angle plus either of this angle or that one will add up to 180 degrees so that's what our exterior angle is so let's go back to the question two angles of a triangle have measure 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 we can figure out all of the exterior angles so first of all let's see what's this 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 X plus let's see 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 if I extend this line out like I did in the example i defined what an exterior angle is this exterior angle would be 120 degrees if I were to do it here if I would extend this out right here what would this exterior angle be well 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 C 180 this is going to be supplementary to 55 so 180 minus 55 is one 25 right right 180 minus 60 would be 120 and then it's only 55 so 125 so the three supplementary or three exterior angles of these 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 right none of the exterior angles are equal to 130 degrees problem 43 43 okay 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 this is a 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 an ax polygon 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 cross is 115 125 and the other one was 120 this was for a triangle if you added them up you get 5 Plus 5 10 and then that's 6 you got 360 degrees for that triangle which had kind of strange angles right it wasn't like an equilateral triangle or anything beautiful and it's also the same if I were to draw a rectangle well I didn't want to draw something to not draw a solid rectangle so if I have a rectangle like that what are the exterior angles here where I like it either see I could drop a I could continue this line right here this angle right here is going to be 90 I could continue this I could go either way I could continue this up or I could go that 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 go like that that exterior angle is 90 so once again 90 plus 90 plus 90 plus 9 is 3 sixty degrees so it's a good thing to know is that the sum of the exterior angles of any polygon is actually 360 degrees and maybe we'll prove that in another video for you know a polygon of size with n sides but anyway now that we know that so if 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 that 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 essentially saying what what quadrant what what polygons interior angles add up to 360 degrees and that of course is a quadrilateral my mouth got ahead of me and if you think you know in a quadrilateral you have 90 90 90 90 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 okay all right what is the measure of angle of angle X so this is an exterior angle to the vertex B right so how do we figure this out well there's two there's a kind of a fast way in a slow way and the slow way is to figure out this angle because you know that the sum of the angles are add up to 180 and then you say oh X is going to be 180 minus that right but isn't if you think about it this angle right here 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 right so we know that X one of them that's X 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 60 225 so this is just saying that the angles within the interior angles of a triangle add up to 180 degrees now we also know that Y plus X is e and we could figure out why right now you could subtract 85 from both sides and you would get Y is equal to what if you ate 95 right and then we could figure out at from why because X is the supplement of Y's and then you could say oh X is equal to 180 minus 95 where the ending you to get 85 and that would be fine that didn't take you too long sees the answer but a slightly faster way of saying it with okay y plus 85 is equal to 180 and you also know that y plus X is equal to 180 all right so clearly X is equal to 85 right if if you add 85 to Y you can 180 if you add X to Y you get 180 so X would be 85 that'd be a slightly faster way of thinking about it but either way is fine if you're not under time pressure okay 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 okay so if the measure of an exterior angle for regular so if we if we think about what an extra if 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 angle so I at 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 right 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 right regular polygon all the angles and all the sides are congruent so an equilateral triangle looking something like that would do the trick all right it's a regular polygon all the sides are the same and it's angles are 60 60 and 60 so when they say how many sides is the polygon half well has three it's a triangle and I'm out of time I'll see you in the next video