Since quadrilateral PQRS is inscribed in circle O, opposite angles are supplementary. This means that the sum of angles P and R must equal 180°.
Given:
- Angle P = (x + 16)°
- Angle R = (6x - 4)°
Setting up the equation:
\[ (x + 16) + (6x - 4) = 180 \]
Combine like terms:
\[ x + 6x + 16 - 4 = 180 \]
\[ 7x + 12 = 180 \]
Subtract 12 from both sides:
\[ 7x = 168 \]
Now, divide by 7:
\[ x = 24 \]
Now, we can find angle P and angle R.
-
Angle P: \[ P = x + 16 = 24 + 16 = 40° \]
-
Angle R: \[ R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140° \]
The angles of the quadrilateral PQRS are as follows:
- Angle P = 40°
- Angle Q = unknown (but can be calculated)
- Angle R = 140°
- Angle S = unknown (but can be calculated)
Since we only need to find the measure of the smaller angle, the smaller of the two angles calculated is 40°.
So the measure of the smaller angle is:
\[ \boxed{40°} \]
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