The sum of the three angles in any triangle sum to 180 degrees. The importance of this fact in Geometry cannot be emphasized enough. The triangle angle sum theorem is used in almost every missing angle problem, in the exterior angle theorem, and in the polygon angle sum formula.
One of the most important properties of triangles that we use all over Geometry is this triangle, angle, sum. What this is talking about is the 3 angles in a triangle and what they're always going to add up to. We're going to look at this two different ways, the first one using a physical model of a triangle secondly we're going to walk over here and we're going to look at this triangle and prove it on the board using some Algebra. So let's start off with this triangle, what's a way that I can determine what these 3 angles of the scaling triangle have to add up to? Well, what I could do, is that I could tear off each of these 3 angles. So here I've got 1 piece and I'm going tear off another piece and what I've done is I've created 3 different angles. So what I'm going to do, is I'm going too take these 3 angles which I've already precut out and taped and I'm going to tape them on the board to see if we can figure anything out. So here's angle 1 and I'm going to tape on the board and I'm going to grab angle 2 and so you can make a difference between the different angles I'm going to leave a little bit of space here, so here's angle 2, so that way you can see that there's a little bit of a difference and last angle 3, and again leaving a little bit of space so you can see that we have 3 different angles and they form a straight line. We know that a straight line is always 180 degrees, so what we can say is that angle 1 plus angle 2 plus angle 3 sum to 180 degrees. But does this always work? It will work for any triangle. So let's move over to the Algebraic explanation of this, right here we have a triangle with 3 angles labeled 1, 2 and 3, and for this proof I'm going to draw in 3 auxiliary lines. The first auxiliary line or helping line is going to be parallel to this base containing 1 and 3. Now to extend our 2 other auxiliary lines I'm going to think of these as transversals that is a line that intersects 2 parallel lines. So we've got 1 transversal right there and we have our second transversal right here. I'm also going to add in 2 more angles, angle 4 and angle 5. So what can we say about 4, 2 and 5, well it's pretty clear that they are all forming one line. So I'm going to say that angle 4 plus angle 2 plus angle 5 sum to 180 degrees. If I compare my 2 equations, one equation has our 3 angles 1, 2, 3 so this is where we want to end up, the other one has 4 and 5 instead of 1 and 3. So how can we rectify this problem, well I see that angel 4 and angle 1 are alternate interior angles. So what I'm going to do, I'm going to say angle 4 is congruent to angle 1 by alternate interior angles. By the same token we can say that 5 and 3 are alternate interior angles as well. Which means angle 5 is congruent to angle 3 by alternate interior angles. So what I'm going to is I'm going to substitute in for 4 with angle 1 and in for angle 5 angle 3. So I'm going to write angle 1 plus angle 2 plus angle 3 sum to 180 degrees. So we've proved it two ways, one by tearing up the 3 pieces of a triangle and seeing that they form a line and secondly using this Algebraic proof where we drew this parallel line that helped us prove that the 3 angles in a triangle will sum to 180 degrees.