Classifying triangles

What I want to do in this video is talk about the two main ways that triangles are categorized. The first way is based on whether or not the triangle has equal sides, or at least a few equal sides. Then the other way is based on the measure of the angles of the triangle. So the first categorization right here, and all of these are based on whether or not the triangle has equal sides, is scalene. And a scalene triangle is a triangle where none of the sides are equal. So for example, if I have a triangle like this, where this side has length 3, this side has length 4, and this side has length 5, then this is going to be a scalene triangle. None of the sides have an equal length. Now an isosceles triangle is a triangle where at least two of the sides have equal lengths. So for example, this would be an isosceles triangle. Maybe this has length 3, this has length 3, and this has length 2. Notice, this side and this side are equal. So it meets the constraint of at least two of the three sides are have the same length. Now an equilateral triangle, you might imagine, and you'd be right, is a triangle where all three sides have the same length. So for example, this would be an equilateral triangle. And let's say that this has side 2, 2, and 2. Or if I have a triangle like this where it's 3, 3, and 3. Any triangle where all three sides have the same length is going to be equilateral. Now you might say, well Sal, didn't you just say that an isosceles triangle is a triangle has at least two sides being equal. Wouldn't an equilateral triangle be a special case of an isosceles triangle? And I would say yes, you're absolutely right. An equilateral triangle has all three sides equal, so it meets the constraints for an isosceles. So by that definition, all equilateral triangles are also isosceles triangles. But not all isosceles triangles are equilateral. So for example, this one right over here, this isosceles triangle, clearly not equilateral. All three sides are not the same. Only two are. But both of these equilateral triangles meet the constraint that at least two of the sides are equal. Now down here, we're going to classify based on angles. An acute triangle is a triangle where all of the angles are less than 90 degrees. So for example, a triangle like this-- maybe this is 60, let me draw a little bit bigger so I can draw the angle measures. That's a little bit less. I want to make it a little bit more obvious. So let's say a triangle like this. If this angle is 60 degrees, maybe this one right over here is 59 degrees. And then this angle right over here is 61 degrees. Notice they all add up to 180 degrees. This would be an acute triangle. Notice all of the angles are less than 90 degrees. A right triangle is a triangle that has one angle that is exactly 90 degrees. So for example, this right over here would be a right triangle. Maybe this angle or this angle is one that's 90 degrees. And the normal way that this is specified, people wouldn't just do the traditional angle measure and write 90 degrees here. They would draw the angle like this. They would put a little, the edge of a box-looking thing. And that tells you that this angle right over here is 90 degrees. And because this triangle has a 90 degree angle, and it could only have one 90 degree angle, this is a right triangle. So that is equal to 90 degrees. Now you could imagine an obtuse triangle, based on the idea that an obtuse angle is larger than 90 degrees, an obtuse triangle is a triangle that has one angle that is larger than 90 degrees. So let's say that you have a triangle that looks like this. Maybe this is 120 degrees. And then let's see, let me make sure that this would make sense. Maybe this is 25 degrees. Or maybe that is 35 degrees. And this is 25 degrees. Notice, they still add up to 180, or at least they should. 25 plus 35 is 60, plus 120, is 180 degrees. But the important point here is that we have an angle that is a larger, that is greater, than 90 degrees. Now, you might be asking yourself, hey Sal, can a triangle be multiple of these things. Can it be a right scalene triangle? Absolutely, you could have a right scalene triangle. In this situation right over here, actually a 3, 4, 5 triangle, a triangle that has lengths of 3, 4, and 5 actually is a right triangle. And this right over here would be a 90 degree angle. You could have an equilateral acute triangle. In fact, all equilateral triangles, because all of the angles are exactly 60 degrees, all equilateral triangles are actually acute. So there's multiple combinations that you could have between these situations and these situations right over here.