To solve for the measures of angles P and R in the inscribed quadrilateral PQRS, we start by using the property that opposite angles of a cyclic quadrilateral (inscribed in a circle) add up to 180 degrees.
Given:
- Angle P = \( (x + 28)^\circ \)
- Angle R = \( (3x - 12)^\circ \)
According to the cyclic quadrilateral property, we can write:
\[ \text{Angle P} + \text{Angle R} = 180^\circ \]
Substituting the values of angles P and R into the equation:
\[ (x + 28) + (3x - 12) = 180 \]
Now, simplify the left side:
\[ x + 28 + 3x - 12 = 180 \] \[ 4x + 16 = 180 \]
Next, we solve for \( x \) by first isolating the term with \( x \):
\[ 4x = 180 - 16 \] \[ 4x = 164 \]
Now, divide by 4:
\[ x = \frac{164}{4} = 41 \]
Now that we have found \( x = 41 \), we can substitute \( x \) back into the expression for angle R to find its measure:
\[ \text{Angle R} = 3x - 12 \] \[ \text{Angle R} = 3(41) - 12 \] \[ \text{Angle R} = 123 - 12 \] \[ \text{Angle R} = 111^\circ \]
Therefore, the measure of angle R is:
\[ \boxed{111 \text{ degrees}} \]