In a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary. This means that the sum of the measures of opposite angles equals 180°.
Given:
- Angle P = \( (x + 16)° \)
- Angle R = \( (6x - 4)° \)
Setting up the equation for these two opposite angles:
\[ (x + 16) + (6x - 4) = 180 \]
Simplifying this equation:
\[ x + 16 + 6x - 4 = 180 \]
\[ 7x + 12 = 180 \]
Subtracting 12 from both sides:
\[ 7x = 168 \]
Dividing by 7:
\[ x = 24 \]
Now substituting \( x \) back into the expressions for angles P and R:
For angle P:
\[ P = x + 16 = 24 + 16 = 40° \]
For angle R:
\[ R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140° \]
Now, since angles S and Q are also opposite to angles P and R, they must also be supplementary:
\[ S + Q = 180° \]
To find the measures of angles Q and S, we would typically need more information, but we can focus on finding the smaller angle.
We have:
- Angle P = 40°
- Angle R = 140°
The smaller angle is \( P = 40° \).
Thus, the measure of the smaller angle is \( \boxed{40°} \).
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