In the diagram, what is the value of x?
The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. The unmarked angles in the middle of the diagram (where the straight lines cross) are the same in the left-hand triangle and in the right-hand triangle. So, if the angles in the left-hand trangle are to sum to 180 degrees (as in the right--hand triangle) it must be true that x+53 = 68+31. This means that x must equal 46.