Isosceles & equilateral triangles problems

Let's do some example problems using our newly acquired knowledge of isosceles and equilateral triangles. So over here, I have kind of a triangle within a triangle. And we need to figure out this orange angle right over here and this blue angle right over here. And we know that side AB or segment AB is equal to segment BC, which is equal to segment CD. Or we could also call that DC. So first of all, we see that triangle ABC is isosceles. And because it's isosceles, the two base angles are going to be congruent. This is one leg. This is the other leg right over there. So the two base angles are going to be congruent. So we know that this angle right over here is also 31 degrees. Well, if we know two of the angles in a triangle, we can always figure out the third angle. They have to add up to 180 degrees. So we could say 31 degrees plus 31 degrees plus the measure of angle ABC is equal to 180 degrees. You can subtract 62. This right here is 62 degrees. You subtract 62 from both sides. You get the measure of angle ABC is equal to-- let's see. 180 minus 60 would be 120. You subtract another 2. You get 118 degrees. So this angle right over here is 118 degrees. Let me just write it like this. This is 118 degrees. Well, this angle right over here is supplementary to that 118 degrees. So that angle plus 118 is going to be equal to 180. We already know that that's 62 degrees. 62 plus 118 is 180. So this right over here is 62 degrees. Now, this angle is one of the base angles for triangle BCD. I didn't draw it that way, but this side and this side are congruent. BC has the same length as CD. Those are the two legs of an isosceles triangle. You can kind of imagine it was turned upside down. This is the vertex. This is one base angle. This is the other base angle. Well, the base angles are going to be congruent. So this is going to be 62 degrees, as well. And then finally, if you want to figure out this blue angle, the blue angle plus these two 62-degree angles are going to have to add up to 180 degrees. So you get 62 plus 62 plus the blue angle, which is the measure of angle BCD, is going to have to be equal to 180 degrees. These two characters-- let's see. 62 plus 62 is 124. You subtract 124 from both sides. You get the measure of angle BCD is equal to-- let's see. If you subtract 120, you get 60, and then you have to subtract another 4. So you get 56 degrees. So this is equal to 56 degrees. And we're done. Now, we could do either of these. Let's do this one right over here. So what is the measure of angle ABE? So they haven't even drawn segment BE here. So let me draw that for us. And so we have to figure out the measure of angle ABE. So we have a bunch of congruent segments here. And in particular, we see that triangle ABD, all of its sides are equal. So it's an equilateral triangle, which means all of the angles are equal. And if all of the angles are equal in a triangle, they all have to be 60 degrees. So all of these characters are going to be 60 degrees. Well, that's part of angle ABE, but we have to figure out this other part right over here. And to do that, we can see that we're actually dealing with an isosceles triangle kind of tipped over to the left. This is the vertex angle. This is one base angle. This is the other base angle. And the vertex angle right here is 90 degrees. And once again, we know it's isosceles because this side, segment BD, is equal to segment DE. And once again, these two angles plus this angle right over here are going to have to add up to 180 degrees. So you call that an x. You call that an x. You've got x plus x plus 90 is going to be 180 degrees. So you get 2x plus-- let me just write it out. Don't want to skip steps here. We have x plus x plus 90 is going to be equal to 180 degrees. x plus x is the same thing as 2x, plus 90 is equal to 180. And then we can subtract 90 from both sides. You get 2x is equal to 90. Or divide both sides by 2. You get x is equal to 45 degrees. And then we're done because angle ABE is going to be equal to the 60 degrees plus the 45 degrees. So it's going to be this whole angle, which is what we care about. Angle ABE is going to be 60 plus 45, which is 105 degrees. And now we have this last problem over here. This one looks a little bit simpler. I have an isosceles triangle. This leg is equal to that leg. This is the vertex angle. I have to figure out B. And the trick here is like, wait, how do I figure out one side of a triangle if I only know one other side? Don't I need to know two other sides? And we'll do it the exact same way we just did that second part of that problem. If this is an isosceles triangle, which we know it is, then this angle is going to be equal to that angle there. And so if we call this x, then this is x as well. And we get x plus x plus 36 degrees is equal to 180. The two x's, when you add them up, you get 2x. And then-- I won't skip steps here. 2x plus 36 is equal to 180. Subtract 36 from both sides, we get 2x-- that 2 looks a little bit funny. We get 2x is equal to-- 180 minus 30 is 150. And then you want to subtract another 6 from 150, gets us to 144. Did I do that right? 180 minus 30 is 150, yep, 144. Divide both sides by 2. You get x is equal to 72 degrees. So this is equal to 72 degrees. And we are done.