You make use of the relationships between angles in a circle, and the relationships between angles in a triangle. You are expected to know or have ready reference to the necessary relationships.
Upper Left. .. x is the base angle of an isosceles triangle whose apex angle is twice 63°. .. x = 27°
Upper Right. .. x is supplementary to 73°. .. x = 107°
Lower Left. .. x is complementary to 50°. .. x = 40°
Lower Right. .. x is supplementary to twice the angle partially shown at the right side of the image. (The central angle between the radii to the tangent points is twice the angle partially shown. The angle to the radius at each point of tangency is 90°, and the sum of the 4 angles in the quadrilateral is 360°. Hence the relationship described in the first sentence.)
