Yes, x is 60 degrees because the other two angles in the triangle sum to 120 degrees. That means that x subtends and arc in the circle of 2(60)= 120 degrees. If you were to draw the radii from the points where that angle crosses the circle to the center, you would have an angle at the center of the circle of 120 degrees. The angles where those radii meet the tangents from y are, of course, 90 degrees. Thus, you would have a quadrilateral with angles of 90, 90, 120, and y. Since the angles in a quadrilateral add to 360 degrees, you have 90+ 90+ 120+ y= 360 so y= 360- 300= 60 degrees.
It's hard to believe you don't know what the picture for the second problem looks like. Draw a circle. Mark two points on the circle, B and C. Let A be any point outside the circle and draw lines AB and AC. To find the degree measure of the two arcs (I would NOT say "angles") a similar argument to the first problem works. If you draw the radii from B and C to the center of the circle, you get a quadrilateral in which one angle, the one at A, is 116 degrees and two others, at B and C, are 90 degrees. Again, the sum of angles in a quadrilateral is 360 degrees so x+ 116+ 90+ 90= 360 where x is the measure of the angle at the center of the circle and so the measure of the arc with endpoints B and C. 360 minus that measure is the measure of the other arc.