To solve for the measure of angle R in quadrilateral PQRS inscribed in circle O, we will use the property of cyclic quadrilaterals. Specifically, the opposite angles of a cyclic quadrilateral sum up to \(180^\circ\).
In this case, angles P and R are opposite angles. According to the problem:
- Angle P: \( P = x + 10 \)
- Angle R: \( R = 5x - 28 \)
According to the property of cyclic quadrilaterals, we have: \[ P + R = 180^\circ \]
Substituting the expressions for angles P and R into the equation: \[ (x + 10) + (5x - 28) = 180 \]
Now, combine like terms: \[ x + 10 + 5x - 28 = 180 \] \[ 6x - 18 = 180 \]
Next, we will add 18 to both sides: \[ 6x = 198 \]
Now, divide both sides by 6 to solve for \(x\): \[ x = 33 \]
Now that we have the value of \(x\), we can find the measure of angle R: \[ R = 5x - 28 \] Substituting the value of \(x\) into the equation for angle R: \[ R = 5(33) - 28 \] \[ R = 165 - 28 \] \[ R = 137 \]
Thus, the measure of angle R is: \[ \boxed{137^\circ} \]