We have three problems that are basically the same, so let's try to come up with a general formula.
As you've observed in the previous problems I've helped you with, the angle cut off by a tangent and a chord is half that of the arc they cut off. So, in the first two problems, if we let the angle we're given be [tex]a[/tex], we have [tex]x = 2a[/tex].
Then, by the straight-line angle rule (angles that together form a straight line sum to 180 degrees), [tex]z = 180 - a[/tex]. Applying the first rule again, [tex]y = 2z = 360 - 2a[/tex].
Applying these formulas, for the first question, [tex]x = 115 \cdot 2 = 230[/tex], [tex]y = 180 - x = 65[/tex], and [tex]z = 360 - 2a = 130[/tex].
For the second problem, [tex]x = 180[/tex], [tex]z = 90[/tex], and [tex]y = 180[/tex].
For the third, the variables are positioned differently, so we replace [tex]x[/tex] with [tex]z[/tex] (because [tex]z[/tex] is now the arc cut off by our given angle) and [tex]z[/tex] with [tex]x[/tex].
So, [tex]z = 2 \cdot 134 = 268[/tex], [tex]x = 180 - 134 = 46[/tex], and [tex]y = 2x = 92[/tex].
