In a cyclic quadrilateral, the opposite angles are supplementary, which means that their measures add up to \(180^\circ\).
Given:
- Angle \(P\) = \(x + 16\)
- Angle \(R\) = \(6x - 4\)
Setting up the equation based on the property of cyclic quadrilaterals: \[ (x + 16) + (6x - 4) = 180 \]
Simplifying this equation: \[ x + 16 + 6x - 4 = 180 \] \[ 7x + 12 = 180 \]
Now, subtract 12 from both sides: \[ 7x = 168 \]
Dividing by 7: \[ x = 24 \]
Now, substituting \(x\) back into the expressions for the angles \(P\) and \(R\):
-
Angle \(P\): \[ P = x + 16 = 24 + 16 = 40^\circ \]
-
Angle \(R\): \[ R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140^\circ \]
Therefore, the measures of angle \(P\) and angle \(R\) are \(40^\circ\) and \(140^\circ\), respectively.
To find the smaller angle: The smaller angle between \(P\) and \(R\) is: \[ \text{Smaller angle} = 40^\circ \]
Thus, the measure of the smaller angle is \( \boxed{40^\circ} \).