For a quadrilateral inscribed in a circle, opposite angles are supplementary, meaning their measures add up to 180 degrees. In this case, angle \( P \) measures \( (x + 16) \) and angle \( R \) measures \( (6x - 4) \).
We can set up the following equation:
\[ (x + 16) + (6x - 4) = 180 \]
Now, simplify and solve for \( x \):
\[ x + 16 + 6x - 4 = 180 \] \[ 7x + 12 = 180 \] \[ 7x = 180 - 12 \] \[ 7x = 168 \] \[ x = \frac{168}{7} = 24 \]
Now that we have \( x \), we can find the measures of angles \( P \) and \( R \):
\[ \text{Angle } P = x + 16 = 24 + 16 = 40 \] \[ \text{Angle } R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140 \]
Since we are looking for the smaller angle, we see that \( 40 \) degrees is the smaller of the two angles.
Thus, the measure of the smaller angle is:
\[ \boxed{40} \]
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