If the measure of the tangent-chord angle is 74 degrees, then what is the measure of the intercepted arc inside the angle?

An angle formed by a tangent line and a chord is a special case of an inscribed angle. The formula is the same however. 

The measure of an inscribed angle is half that of the measure of the interecpted arc. That means the measure of the subtended arc is twice that of the measure of the inscribed angle or 2*74 = 148°.