Finding angles in isosceles triangles

The measure of two angles of an isosceles triangle are 3x plus 5 degrees, we'll 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 and like that. And actually, let me draw a couple of them just because we want to think about all of the different possibilities here. So we know, from what we know about isosceles triangles, that the base angles are going to be congruent. So 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 degrees. The other possibility is that this is describing both base angles, in which case, they would be equal. So maybe this one is 3x plus 5, and maybe this one over here is x plus 16. 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. 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 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 are going to have to add up to 180 degrees. So we get 3x plus 5 plus 3x plus 5 plus x plus 16 is going to be equal to 180 degrees. We have 3x. Let's just add up. You have 3x plus 3x, which gives you 6x, plus another x gives you 7x. And then you have 5 plus 5, which is 10, plus 16 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 20 is 160, minus another 6 is 154. You have 7x is equal to 154. And let's see how many times-- 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 degrees in this the first scenario. Now let's think about the second scenario over here. 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 by 2, and you get x is equal to 11/2. 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 16 plus 3x plus 5. 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. And then you have 16 plus 16 is 32. 32 plus 5 is 37. Plus 37 is equal to 180 degrees. Subtract 37 from both sides, and we get 5x is equal to 180 minus 30 is 150. So that gets us to 143. So it's not going to divide nicely. Divide both sides by 5, you get x is equal 143/5, which we can just leave as an improper fraction. You could write it as a mixed number or however else you might want to write it. And we're done. These are the three possible values of x, given the information that they gave us right up there.