In a cyclic quadrilateral (one that is inscribed in a circle), the opposite angles are supplementary. This means that the sum of the measures of angles P and R is equal to 180°.
Given:
- Angle P = \(x + 16\)
- Angle R = \(6x - 4\)
We can set up the equation based on the property of opposite angles in a cyclic quadrilateral:
\[ (x + 16) + (6x - 4) = 180 \]
Now, let's simplify this equation:
\[ x + 16 + 6x - 4 = 180 \] \[ 7x + 12 = 180 \]
Next, subtract 12 from both sides:
\[ 7x = 180 - 12 \] \[ 7x = 168 \]
Now, divide both sides by 7 to solve for \(x\):
\[ x = \frac{168}{7} = 24 \]
Now that we have the value of \(x\), we can find the measures of angles P and R.
For angle P: \[ P = x + 16 = 24 + 16 = 40^\circ \]
For angle R: \[ R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140^\circ \]
Therefore, the measures of angles P and R are:
- Angle P = 40°
- Angle R = 140°
To find the smaller angle, we look at the measures of the angles:
- The smaller angle of the two is \(40^\circ\).
Thus, the answer is \( \boxed{40^\circ} \).
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