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CCSS.Math: ,

let's do some example problems using our newly acquired knowledge of isosceles and equilateral triangles so over here I have a 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 a B is equal to our segment a B 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 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 if we call we could say 31 degrees plus 31 degrees plus the measure of angle ABC ABC is equal to 180 degrees you can subtract 60 to this right here 62 degrees you subtract 62 from both sides you get the measure of angle ABC is equal to let's see 180 minus sixty would be 120 you subtract another two you get 118 degrees so this angle right over here is one hundred 118 degrees let me just write like this well this is 118 degrees well this angle right over here 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 60 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 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 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 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 B C D is equal to let's see if you subtract 120 you get 60 and then you have to subtract another 4 so you get 50 you get 56 degrees so this is equal to 56 degrees and we're done now let's we could do either of these let's do this one right over here so what is the measure of angle a b e so they haven't even drawn segment to be he here so let me draw that for us and so we have to figure out the measure of angle a b e 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 they all have to be 60 degrees so that all of these characters are going to be 60 degrees well that's part of angle a be e 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 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 notice isosceles because this side segment BD is equal to segment de and once again these two angles Plus this angle right over here is going to have to add up to 90 degrees so you call that an X you call that an X you get X plus X plus 90 is going to be 180 degrees so you get 2x plus let me just write it out I 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 X is equal to 45 degrees and then we're done because angle a B E 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 a B E is going to be a 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 and I know 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 or not 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 we get X plus X plus 36 degrees plus 36 is equal to 180 the two X's when you add them up you get 2x and then I'll just I won't skip the steps here 2x plus 36 is equal to 180 subtract 36 from both sides we get 2x 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