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CCSS.Math: ,

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 is 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 well 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 if I keep going on and on forever in the same directions then now all of a sudden I have an orange 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 can 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 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 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 then that means any of of two parallel lines 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 you have this angle right over here these two angles are vertical so if this has measure X then this one must have as your 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 and the bottom parallel line what angle does it 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 angle or it's going to have measure Y as well so now we're really at the homestretch of our proof because we will see that the measure we have this angle in this angle this has measure angle X this has measure Z they're both adjacent angles if we take 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 what's this measure of this white angle right over there well it's going to be X plus Z X plus Z and that angle is supplementary is supplementary to this angle right over here that has measure Y so the measure of X measure 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 one 180 degrees and we are done