It would help if they put some labels on the picture.
y looks easy. We do that first. The subtended angle is half the central angle, so half the arc measure. So
Answer: y = 23 degrees
OK, z next. Clearly the figure isn't to scale; compare that 34 degrees and the 46 degrees. Anyway, it's 360 degrees around the circle, so that leaves
z = 360 - (160+46+34) = 120
Answer: z = 120 degrees
x is not really determined unless that we're supposed to assume that where the left side touches the circle is a tangent. Let's call that point T. Let's call X and Y the vertices near x and y on the figure. We'll call the center C.
Clearly we can extend the base XY of the triangle, call the endpoint X and connect T to it. So unless we have the tangent constraint or something else, x isn't determined.
So we'll assume XT is tangent to the circle. Angle TYX, the vertex by y, is half of 46+34, so TYX=40 degrees.
TCY is an isosceles triangle, two radii for sides. Angle TCY=160 degrees. so CYT=CTY=10 degrees.
The radius and the tangent make a right angle so that makes angle XTY=90+10=100.
We have XTY=100, TYX=40 so that leaves TXY=40, and the triangle in the figure is isosceles.
Answer: x = 40 degrees
