Theorems concerning triangle properties
I've drawn an arbitrary triangle right over here. And I've labeled the measures of the interior angles. The measure of this angle is x. This one's y. This one is z. And what I want to prove is that the sum of the measures of the interior angles of a triangle, that x plus y plus z is equal to 180 degrees. And the way that I'm going to do it is using our knowledge of parallel lines, or transversals of parallel lines, and corresponding angles. And to do that, I'm going to extend each of these sides of the triangle, which right now are line segments, but extend them into lines. So this side down here, if I keep going on and on forever in the same directions, then now all of a sudden I have an orange line. And what I want to do is construct another line that is parallel to the orange line that goes through this vertex of the triangle right over here. And I can always do that. I could just start from this point, and go in the same direction as this line, and I will never intersect. I'm not getting any closer or further away from that line. So I'm never going to intersect that line. So these two lines right over here are parallel. This is parallel to that. Now I'm going to go to the other two sides of my original triangle and extend them into lines. So I'm going to extend this one into a line. So, do that as neatly as I can. So I'm going to extend that into a line. And you see that this is clearly a transversal of these two parallel lines. Now if we have a transversal here of two parallel lines, then we must have some corresponding angles. And we see that this angle is formed when the transversal intersects the bottom orange line. Well what's the corresponding angle when the transversal intersects this top blue line? What's the angle on the top right of the intersection? Angle on the top right of the intersection must also be x. The other thing that pops out at you, is there's another vertical angle with x, another angle that must be equivalent. On the opposite side of this intersection, you have this angle right over here. These two angles are vertical. So if this has measure x, then this one must have measure x as well. Let's do the same thing with the last side of the triangle that we have not extended into a line yet. So let's do that. So if we take this one. So we just keep going. So it becomes a line. So now it becomes a transversal of the two parallel lines just like the magenta line did. And we say, hey look this angle y right over here, this angle is formed from the intersection of the transversal on the bottom parallel line. What angle to correspond to up here? Well this is kind of on the left side of the intersection. It corresponds to this angle right over here, where the green line, the green transversal intersects the blue parallel line. Well what angle is vertical to it? Well, this angle. So this is going to have measure y as well. So now we're really at the home stretch of our proof because we will see that the measure-- we have this angle and this angle. This has measure angle x. This has measure z. They're both adjacent angles. If we take the two outer rays that form the angle, and we think about this angle right over here, what's this measure of this wide angle right over there? Well, it's going to be x plus z. And that angle is supplementary to this angle right over here that has measure y. So the measure of x-- the measure of this wide angle, which is x plus z, plus the measure of this magenta angle, which is y, must be equal to 180 degrees because these two angles are supplementary. So x-- so the measure of the wide angle, x plus z, plus the measure of the magenta angle, which is supplementary to the wide angle, it must be equal to 180 degrees because they are supplementary. Well we could just reorder this if we want to put in alphabetical order. But we've just completed our proof. The measure of the interior angles of the triangle, x plus z plus y. We could write this as x plus y plus z if the lack of alphabetical order is making you uncomfortable. We could just rewrite this as x plus y plus z is equal to 180 degrees. And we are done.