To find the value of \( w \) for the angle \( (8w + 17)^\circ \) in a regular decagon, we first need to calculate the measure of each interior angle of a regular decagon.
The formula for the measure of each interior angle of a regular polygon is given by:
\[ \text{Interior Angle} = \frac{(n-2) \times 180}{n} \]
where \( n \) is the number of sides. For a decagon, \( n = 10 \):
\[ \text{Interior Angle} = \frac{(10-2) \times 180}{10} = \frac{8 \times 180}{10} = \frac{1440}{10} = 144^\circ \]
Now we set the equation for the angle \( (8w + 17)^\circ \) equal to \( 144^\circ \):
\[ 8w + 17 = 144 \]
Next, we will solve for \( w \):
- Subtract \( 17 \) from both sides:
\[ 8w = 144 - 17 \]
\[ 8w = 127 \]
- Now, divide both sides by \( 8 \):
\[ w = \frac{127}{8} \]
Calculating \( \frac{127}{8} \):
\[ w = 15.875 \]
Rounding this to the nearest whole number gives:
\[ w \approx 16 \]
Thus, the value of \( w \) is \( \boxed{16} \).
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