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Proofs concerning equilateral triangles

CCSS.Math:

Video transcript

what we've got over here is a triangle where all three sides have the same length or all three sides are congruent to each other and a triangle like this we call equilateral this is an equilateral triangle equilateral triangle now what I want to do is prove that if all three sides are the same then we know that all three angles are going to have the same measure so let's think how we can do this well first of all we could just look at we know that a B is equal to AC so let's just pretend that we don't even know that this also happens to be equal to BC and we know for if the four isosceles triangles if two legs if two legs have the same length then the base angles have the same length and then so let's write this down we know that triangle ABC is going to be congruent to angle a/c B let me write this down we know angle a BC is congruent to angle ACB because because so maybe this is my statement right over here statement statement and then we have reason and we the reason here and I'll write it in just kind of shorthand is that their base angles of I guess you could say nice oscillate because we know that this side is equal to that side now obviously this is an equilateral all of the sides are equal but the fact that these two legs are equal so that the base angles are equal so we say two legs two legs equal imply base angles base angles are going to be equal and that just comes from what we actually did in the last video with isosceles triangles but we can also view this triangle the other way we could also say that maybe maybe this angle over here is the vertex angle maybe these two are the base angles because then you have a situation where this side and this side are congruent to each other and then that angle and that angle are going to be base angles so you could say angle C a B angle C a B is going to be congruent so angle C a B is going to be congruent to angle ABC to angle ABC ABC ABC really for the same reason we're now looking at different different legs here and different base angles this would now be the base in this example we kind of you could imagine turning an isosceles triangle on its side but it's the exact same logic so let's just review what I talked about these two sides are equal which imply these two base angles are equal these two sides being equal implied these two base angles are equal well if ABC is congruent to ACB and it's congruent to CE a/b then all of these angles are congruent to each other so they are all so then we get we get angle ABC is congruent to angle ACB which is congruent to angle c a.b and that pretty much gives us all of the angles so if you have an equilateral triangle it is actually an equiangular triangle as well all of the angles are going to be the same and you actually know what that measure is if you have three things of the same so let's call that X X X and they add up to 180 you get X plus X plus X is equal to 180 or 3x is equal to 180 divide both sides by 3 you get X is equal to 60 degrees so in an equilateral triangle not only are they all the same angles but they're all equal to exactly they're all 60-degree they are all 60 degree angles now let's think about it the other way around let's say I have a triangle let's say we've got ourselves a triangle where all of the angles are the same all of the angles so let's say that's point X point y and point Z and we know that all the angles are the same so we know that this angle is congruent to this angle is congruent to that angle so what we show what we showed in the last video on isosceles triangles is that if two base angles are the same then the corresponding legs are also going to be the same so we know for example that Y X is congruent to Y Z we know Y is congruent to Y to Y Z and we know that because because they're the base angles are congruent base angles base angles congruent now we also know we also know that Y Z so I'll rewrite Y Z is congruent to X Z so we also know that Y Z is congruent to XZ by the same argument but here we're dealing with different base angles so now once again you can view this as almost an isosceles triangle turned on its side this is the vertex angle right over here these are the two base angles this would be the base now and we know that because the base angle these two base angles are congruent so by the same logic over in this first case the base angles was where this angle and that angle and the second case the base angles are that angle in that angle and actually let me write it down the base angles in this first case let me do that same magenta our angle X Y or Y X Z so this angle Y XZ is congruent to angle Y Z X Y Z X that was in the first case these are the base angles so based on the proof we saw in the last video that implies these sides are congruent here we have these two base angles so here we're saying angle let me do that green angle X Y Z X Y Z is congruent to angle yxz Y X Z and so that implies that these two guys right over here are congruent well there we've proved it we said that this side Y X is congruent to Y Z and we've shown that Y Z is congruent to XZ so all of the sides are congruent to each other so once again if you have all the angles equal and they're going to have to be 60 degrees then you know that all of the sides are going to be equal as well they're going to be congruent