 Earlier we've talked about equilateral triangles and equi-angular triangles and we have this theorem from the first day's first day's notes A triangle is equilateral if and only if It is equi-angular So here we have an equilateral triangle we can tell because of the congruence marks And so that means each of these three angles must be congruent so if all of the angles add up to 180 and Three are congruent Then 180 divided by 3 is 60 degrees so each of the angles in an equilateral triangle equi-angular are 60 degrees So let's take a look at this theorem a little bit more and and apply it to some problems So right now this triangle Well, what kind of triangle is it? We have congruent angles angle r and angle t and what that tells me is I have an isosceles triangle for sure It's guaranteed to be an isosceles triangle and in particular since these are base angles Then our t is the base and so our s and st would be the congruent legs So let's solve for y and we can only use those pieces of information To solve for y right because they're even though it's tipped on its side Those are the legs of the isosceles triangle So since legs are congruent six y plus five is Equal to 7 y minus 13 So let's solve for y first. Let's subtract 6 y from both sides And then add 13 to both sides So we have y is equal to 18 Now if y is equal to 18 and it says we're to classify this triangle by its sides well, we know for a fact that it's an isosceles triangle because These base angles were congruent However, it's possible remember an isosceles triangle could potentially be an equilateral triangle if all three sides are congruent And so let's take this value of 18 for y and Substitute that value in for each of these three side lengths. So sub in y equals 18 for these three sides So for each of these six y six times 18 plus five that's a hundred and thirteen 78 times 18 minus 13 is also a hundred and thirteen not a surprise there because those are the congruent legs in an isosceles Triangle, but this blue statement five times 18 plus 23 is also equal to a hundred and thirteen And so what that tells me is all three of the sides are congruent to each other and so triangle rst It's not just isosceles. It's actually equilateral