To find \( m\angle SGM \), we can use the properties of tangents and secants with respect to a circle.
Given:
- \( G \) is the point of tangency where line \( GM \) is tangent to circle \( OO \).
- \( S \) is a point on secant \( GS \).
- The measure of arc \( GS \) (denoted \( m\arc{GS} \)) is given as \( 84^\circ \).
According to the Tangent-Secant Theorem, the angle formed by a tangent and a secant drawn from the same external point is equal to half the measure of the intercepted arc. Specifically, this can be applied as:
\[ m\angle SGM = \frac{1}{2} m\arc{GS} \]
Substituting the given value into the formula gives us:
\[ m\angle SGM = \frac{1}{2} \times 84^\circ = \frac{84^\circ}{2} = 42^\circ \]
Thus, the measure of angle \( SGM \) is:
\[ \boxed{42^\circ} \]