Theorems concerning triangle properties
What I want to do now is just a series of problems that really make sure that we know what we're doing 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 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 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 the sum of angles and a triangle equaling 180, or this one doesn't have parallel lines on it. But you might see some with parallel lines and supplementary lines and complementary lines-- is to just fill in everything that you can figure out, and one way or another, you probably would 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 2 of the angles. And if you have 2 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 could 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 4 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 hard to figure out. So let me write it like this. Let me write x is equal to 66 degrees. Well if we know this angle right over here, if we know the measure of this angle is 66 degrees, we know that that angle 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 2 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 2 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, y is an 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 180 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. 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 just reason it through yourself just with the sum of the measures of the angles inside of a triangle add up to 180 degrees, and then you have a supplementary angles right over here. Or you could just say, look, if I have the exterior angles right over here, it's equal to the sum of the remote interior angles. That's just a little terminology you could 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 were given this angle in the beginning. Now we just have to figure out this third angle 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 plus 31 is equal to 180 degrees. 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 in this step. So we get z plus, what is this, 145 is equal to 180. Did I do that right? We have a 15, then a 30. Yep, 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.