The triangle sum theorem states that the sum of the measures of angles in a triangle is 180°. So, if a triangle has two angle measures given, it is possible to find the measure of the third by subtracting the two given measures from 180°.
Recall that the sum of a linear pair of angles, which are adjacent angles that lie on the same line, is also 180°. This, in with the triangle sum theorem, can be used to find various angles within a triangle.
The way that you apply the triangle angle sum is by finding missing angles in triangles. So in this problem we’re looking at two different unknowns x and y and they’re both angles within this triangle. And notice that we actually have three different triangles here. We have a smaller one, we have another slightly larger one and then we have the big triangle where the two triangles are within it.
So how can we find x? To do that we need to go back to our triangle angle sum, which says, if you have three angles of a triangle, they will add up to be 180 degrees. So to find x, what I’m going to do is I’m going to say that x degrees plus the other two angles, 32 degrees plus 41 degrees, must sum to 180 degrees.
So all we have to do is solve this equation for x. So 32 and 41 is 73. So we have x plus 73 degrees equals 180. I’m going subtract 73 degrees from both sides and I find that x must be 107 degrees. So I’m going to write that up here and I’m going to erase the x and write in 107, because now I’m going to use that information to find y.
So there’s two ways to find y and it all depends on which way you feel most comfortable. One way that I would look at this is say that 107 is a linear pair with this angle right here, which is in that smaller triangle. So I’m going to say that this angle right here has to be 73 degrees, because I know those two angles must sum to 180.
So I’m going to do the triangle angle sum again, which says, if I move over to this area right here, that 73 degrees plus 20 degrees plus y degrees is equal to 180 degrees. Again, just combining like terms, we see that 73 and 20 is 93, plus y degrees equals 180.
And last I’m going to subtract 93 degrees from both sides. And I find that y must be 87 degrees. I’m going to write my answer up here and the key thing was noticing that we can use the triangle angle sum to find missing angles.