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Finding angles in isosceles triangles

CCSS.Math:

Video transcript

the measure of two angles of an isosceles triangle or three X plus five degrees will say and X plus 16 degrees find all possible values of X so let's think about this let's draw ourselves an isosceles triangle or two so it's an isosceles triangle like that like that and actually let me draw a couple of them just because we want to think about all of the different possibilities here all of the different possibilities so we know from what we know about isosceles triangles that the base angles are going to be congruent so that's I that angle is going to be equal to that angle that angle is going to be equal to that angle and so what could the 3x plus 5 degrees and the X plus 16 what could they be measures of well maybe this one right over here has a measure of 3x plus 5 degrees and the vertex is the other one so maybe this one up here is the X plus 16 X plus 16 degrees the other possibility the other possibility is that this is describing both base angles in which case they would be equal so maybe this one is 3 X plus 5 and maybe this one over here is X plus 16 X plus 16 and that is and then the final possibility actually we haven't exhausted all of them is if we swap these two if this one is X plus 16 and that one is 3x plus 5 so let me draw ourselves another triangle let me draw ourselves another triangle and obviously swapping these two aren't going to make a difference because they are equal to each other and then we could make that one equal to 3x plus 5 but that's not going to change anything either because they're equal to each other so the last situation is where this angle down here is X plus 16 and this angle up here is X plus is 3x plus 5 this is 3x plus 5 so let's just work through each of these so in this situation if this base angle is 3x plus 5 so is this base angle and then we know that all three of these we're going to have to add up to 1 our 80 degrees so we get 3x plus 5 plus 3x plus 5 plus X plus 16 plus X plus 16 is going to be equal to 180 degrees we have 3x let's just add up the you have 3x plus 3x which is gives you 6x plus another X gives you 7 X 7 X and then you have 5 plus 5 which is 10 plus 16 is equal to 26 is equal to 26 and that is going to be equal to 180 and then we have let's see 180 minus 26 if we subtract 26 from both sides we get 180 minus 180 minus 20 is 160 minus another 6 is 150 for 150 150 4 you have 7x is equal to 154 7x is equal to 154 and let's see how many times this if we divide both sides by 7 7 will go into 140 20 times and then you have another 14 so it looks like it's 22 times so X is equal to 22 is that right 20 times 7 is 140 140 plus 14 is 154 so we have X is equal to 22 in this first 22 degrees in this first scenario now let's think about the second scenario over here now we have now we have these two characters are going to be equal to each other because they're both the base angles so you have 3x plus 5 is equal to X plus 16 well you can subtract X from both sides and so this becomes 2x plus 5 is equal to 16 we can subtract 5 from both sides and you get 2x is equal to 11 and then you can divide both sides you can divide both sides by 2 and you get X is equal to 11 is equal to 11 halves so that is our second scenario and then we do our third scenario right over here if this base angle is X plus 16 then this base angle right over here is also going to be X plus 16 they are congruent and then we can do the same thing that we did for the first scenario all of these angles are going to have to add up to 180 degrees so we have X plus 16 plus X plus 60 mean plus three x plus five plus three X plus five when you add them all together you're going to get 180 degrees now let's add up all the X terms X plus X is 2x plus 3x is 5x so we get 5x 5x and then you have 16 plus 16 is 32 32 plus 5 is 37 plus 37 is equal to 180 degrees is equal to 180 degrees subtract 37 from both sides and we get 5x 5x is equal to 180 minus 30 is 150 so that gives us 2 143 so it's not going to divide nicely divide both sides by 5 you get X is equal to 143 over 5 which we can just leave as an improper fraction you can write it as a mixed number or however else you might want to write it and we're done these are the three these are the three possible values the three possible values of X given the information that they gave us right up there