Triangle exterior angle example

what I want to do now is just a series of problems that really make sure that we know what we're doing with with with parallel lines and triangles and all the rest and what we have right here is a fairly classic problem and what I want to do is I want to figure out just given the information here so obviously I have a triangle here I have another triangle over here we were given some of the angles inside of these triangles given the information over here I want to figure out what the measure of this angle is right over there I need to figure out what that question mark is and so you might want to give a go at it just knowing what you know about the the sums of the measures of the angles inside of a triangle and maybe a little bit of what you know about supplementary angles so you might want to pause it and give it a try yourself and because I'm about to give you the solution so the first thing you might say and this is a general way to think about a lot of these problems where they give you some angles and you have to figure out some other angles based on some of angles in a triangle equaling 180 or this someone doesn't have parallel lines on it but you might see some with parallel lines and and supplementary lines and complementary lines is to just fill in everything that you can figure out and one way or another you probably be able to figure out what this question mark is so the first thing that kind of pops out to me is we have one triangle right over here we have this triangle on the left and on this triangle on the left we're given two of the angles and if you have two of the angles in a triangle you can always figure out the third angle because they're going to add up to 180 degrees so if you call that X we know that X plus 50 plus 64 is going to be equal to 180 degrees or we could say X plus what is this 114 X plus 114 is equal to 180 degrees we can subtract 114 from both sides of this equation and we get X is equal to 180 minus 114 so 80 minus 14 80 minus 10 would be 70 minus another four is 66 so X is 66 degrees now if X is 66 degrees I think you might find that there's another angle that's not too to figure out so let me let me write it like this so X so X is equal to X let me write X is equal to 66 degrees is equal to 66 degrees well if we know this angle right over here if we know the measure of this angle 66 degrees we know that that angle is is supplementary with this angle right over here their outer sides form a straight angle and they are adjacent so if we call this angle right over here Y we know that Y plus X is going to be equal to 180 degrees and we know X is equal to 66 degrees so this is 66 and so we can subtract 66 from both sides and we get Y is equal to these cancel out 180 minus 66 is 114 and that number might look a little familiar to you notice this 114 was the exact same sum of these two angles over here and that's actually a general idea and I'll do it on the side here just to prove it to you if I have let's say that these two angles let's say that the measure of that angle is a the measure of that angle is B the measure of this angle we know is going to be 180 minus a minus B that's this angle right over here and then this angle which is considered to be an exterior angle so in this example why is it exterior angle in this example that is our exterior angle that is going to be supplementary to 180 minus a minus B so this angle Plus what 80 minus a minus B is going to be equal to 180 so if you call this angle Y you would have Y plus 180 minus a minus B is equal to 180 you could subtract 180 from both sides you could add a plus B to both sides so plus a plus B plus a plus B running out of space on the right hand side and then you're left with these cancel out on the left-hand side you're left with Y on the right-hand side is equal to a plus B so this is just a general property you can you can just reason it through yourself just with the sum of the measures of an inside of of the angles inside of a triangle add up to 180 degrees and then you have supplementary angles right over here or you could just say look if I have the exterior angles right over here its equal it's equal to the sum of the remote interior angles that's just a little terminology you'd see there so y is equal to a plus B 114 degrees we've already shown to ourselves is equal to 64 plus 50 degrees but anyway regardless of how we do it if we just reason it out step by step or if we just knew this property from the get-go if we know that Y is equal to 114 degrees and I like to reason it out every time just to make sure I'm not jumping to conclusions so if Y is 114 degrees now we know this angle we know we were given this angle at the beginning now we just have to figure out this third angle in this in this triangle so if we call this Z if we call this question mark is equal to Z we know that Z plus 114 we know that Z plus 114 plus 31 is equal to 180 degrees thus the sums of the measures of the angle inside of a triangle add up to 180 degrees that's the only property we're using this step so we get Z plus what is this 145 145 is equal to 180 I do that right if I we have we have a 15 and then a 30 145 is equal to 180 subtract 145 from both sides of this equation and we are left with Z is equal to 80 minus 45 is equal to 35 so Z is equal to 35 degrees and we are done