In a cyclic quadrilateral (one inscribed in a circle), opposite angles are supplementary. Therefore, we can set up the following equation for angles \( P \) and \( R \):
\[ (x + 16) + (6x - 4) = 180 \]
Now we can simplify this equation:
\[ x + 16 + 6x - 4 = 180 \] \[ 7x + 12 = 180 \] \[ 7x = 180 - 12 \] \[ 7x = 168 \] \[ x = \frac{168}{7} = 24 \]
Now we can find the measures of angles \( P \) and \( R \):
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Angle \( P \): \[ P = x + 16 = 24 + 16 = 40° \]
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Angle \( R \): \[ R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140° \]
Now let's find the other two angles \( Q \) and \( S \). Since angles \( P \) and \( R \) are opposite angles, angles \( Q \) and \( S \) are the other pair of opposite angles in the quadrilateral, and they are also supplementary:
\[ Q + S = 180 \]
Since we know that \( P \) and \( R \) measure 40° and 140° respectively, we can find \( Q \) and \( S \) using the supplementary angles principle:
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Given \( Q + P = 180° \): \[ Q = 180 - 40 = 140° \]
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Given \( S + R = 180° \): \[ S = 180 - 140 = 40° \]
Thus, we have:
- \( P = 40° \)
- \( R = 140° \)
- \( Q = 140° \)
- \( S = 40° \)
The two smaller angles are \( P \) and \( S \) both measuring \( 40° \).
So the measure of the smaller angle is:
\[ \boxed{40°} \]