Find the values of the variables in the parallelogram. the diagram is not drawn to scale. use theorems involving parallel lines and the the sum of the angles of a triangle.

X = 60
y = 52
z = 68

This problem is quite simple if you remember the theorems and don't attempt to solve the angles in alphabetical order. Looking at the figure, the first angle you should immediately realize is y because it's an alternate interior angle to the angle measuring 52 degrees and will have the same value. So we now know that y is 52 degrees.
After you've gotten angle y, you'll then know that there's a triangle where you know 2 of the measurements, which are 68 and 52. And since a triangle has 180 degrees, that will tell you that x is equal to 180 - 68 - 52 = 60. So now you know that x is 60 degrees. And that just leaves you with z. And there's 2 ways you can solve that. One is to realize that opposite vertexes in a parallelogram are equal, so you know it's 68 degrees. The other is to realize that x is an opposite interior angle to the unlabeled angle next to y and that it will have the same value as x. So you then know 2 angles of the triangle, so z = 180 - 52 - 60 = 68, which confirms the value. So the final answers are:
x = 60
y = 52
z = 68