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thought I would do some more example problems involving triangles and so this first one it says the measure of the largest angle in a triangle is 4 times the measure of the second largest angle the smallest angle is 10 degrees what are the measures of all the angles well we know one of them we know it's 10 degrees let's draw an arbitrary triangle right over here so let's say that is our triangle we know that the smallest angle is going to be 10 degrees and I'll just say let's just assume that this right over here is the measure of the smallest angle it's 10 degrees now let's call the second largest angle let's call that let's call that X so the second largest angle let's call that X so this is going to be X and then the first sentence they say the measure of the largest angle in a triangle is 4 times 4 times the measure of the second largest angle so the second largest angle is X 4 times that measure is going to be 4x so the largest angle is going to be is going to be 4x and so the one thing we know about the measures of the angles inside of a triangle is that they add up to 180 degrees so we know that 4 X plus X plus X plus 10 degrees plus 10 degrees is going to be equal to 180 degrees is going to be equal to 180 + 4 X plus X that just gives us 5x and then we have 5 X plus 10 is equal to 180 degrees subtract 10 from both sides you get 5x is equal to 170 and so X is equal to 170 over 5 and let's see it'll go into it what is that 34 times let me verify this so 5 goes into yeah it should be 34 times because it's going to go into twice as many times as 10 would go into it 10 would go into 170 17 times 5 we go into hundred 70 34 times so we could verify it went to 170 five goes into 17 3 times 3 times 5 is 15 subtract you get to bring down the 0 5 goes into 20 four times and then you're not going to have a remain four times five is 20 no remainder so it's 34 times so X is equal to 34 so the second largest angle has a measure of 34 degrees this angle up here is going to be four times that so four times 34 let's see that's going to be 120 degrees plus 16 degrees this is going to be 136 degrees is that right 4 times 4 times 4 is 16 4 times 3 is 120 16 plus 1 or Twenties 136 degrees so we're done the three measures or the three the the sizes of the three angles are 10 degrees 34 degrees and 136 degrees let's do another one so let's see we have a little bit of a drawing here and what I want to do is and we could think about different things we could say let's solve for X I'm assuming that 4x is the measure of this angle 2x is the measure of that angle right over there we can solve for X and then if we know X we can figure out what the actual measures of these angles are assuming that we can figure out X and the other thing that they tell us is that this line over here is parallel to this line over here and was very craftily drawn because it's parallel but they kind of one stops here then one starts up there so the first thing I wondered if they're telling us these two lines are parallel is probably going to be something involved involving transversals or something it might be something involving the other option is something involving triangles and at first you might say wait are is this angle and that angle vertical angles we have to be very careful they are not this is not the same line this this line is parallel to that line it's not it's not this line it's bending right over there so we can't make any type of assumption like that so the interesting thing and I'm not sure if this will lead in the right direction is to just make it clear that these two are these two are part of parallel lines so I could continue this line down like this and then I can can you continue this line up like that and then that starts to look a little bit more like we're used to when we're dealing with parallel lines and then this line segment BC or we could even say line BC if we were to continue it on if we were to continue it on and on even pass D then this is clearly a transversal of those two parallel lines this is clearly a transversal and so if this angle right over here if this angle right over here is 4x it has a corresponding angle half of the or maybe most of the work on all of these is to try to see the parallel lines and see the transversal and see see the one that see the things that might be useful for you so that right there is the transversal these are the parallel lines that's one parallel line that is the other parallel you can almost try to zone out all of the other stuff in the diagram and so if this angle right over here is 4 X it has a corresponding angle on the other where the transversal intersects the other parallel line this right here is its corresponding angle so let me draw it in that same yellow this right over here is a corresponding angle so this will also be this will also be 4x and we see that this angle this angle and this angle this angle that has measure 4x and this measure this angle that measures 2x we see that they're supplementary they're adjacent to each other their outer their outer sides form a straight angle so they're supplementary which means that their measures add up to 180 degrees they kind of form they go all the way around like that if you add the two adjacent angles together so we know that 4x + 2 X needs to be equal to 180 degrees or we get 6x is equal to 180 degrees divide both sides by 6 you get X is equal to 30 or X is equal to well I shouldn't say well X could be 30 and then this angle right over here is 2 times X so it's going to be 60 degrees so this angle right over here is going to be 60 degrees and this angle right over here is this angle right over here is 4 times X so it is it is 120 degrees and we're done