In the diagram below, AB is a tangent at C and XC is a diameter.
mÐ YCB = 58° and mÐ XYZ = 38°.

Find the following: mÐ YXC,  
mÐ YZC, mÐ YCX, mÐ ZCA, 
mÐ ZCX, mÐ ZYC,

 mÐ YXC = 58°      

 (inscribed angle on opposite side of chord CY from the tangent chord angle, Ð YCB)
                                     
mÐ YZC = 58°  (same as above)

mÐ YCX = 90° - 58° = 32° (tangent perpendicular to radius)

mÐ ZYC = 90° - 38° = 52° (Inscribed angle in a semicircle is a right angle)

mÐ ZCA = mÐ ZYC = 52  (tangent chord angle is equal to inscribed angle on the opposite side of the chord. )
                                                 
mÐ ZCX = mÐ XYZ = 38°  (inscribed angles intercepting the same arc are equal.)