For Exercises 5-8, given the measure of an interior angle of a regular polygon, how many sides does each polygon have? 5. 174^circ 6. 156^circ 8. 177.5^circ 7. 165^circ

To determine the number of sides of a regular polygon given the measure of its interior angle, we can use the formula:<br /><br />\[ \text{Number of sides} = \frac{360^{\circ}}{180^{\circ} - \text{Interior angle}} \]<br /><br />Let's apply this formula to each given interior angle:<br /><br />### 5. \(174^{\circ}\)<br /><br />\[ \text{Number of sides} = \frac{360^{\circ}}{180^{\circ} - 174^{\circ}} = \frac{360^{\circ}}{6^{\circ}} = 60 \]<br /><br />So, the polygon has 60 sides.<br /><br />### 6. \(156^{\circ}\)<br /><br />\[ \text{Number of sides} = \frac{360^{\circ}}{180^{\circ} - 156^{\circ}} = \frac{360^{\circ}}{24^{\circ}} = 15 \]<br /><br />So, the polygon has 15 sides.<br /><br />### 7. \(165^{\circ}\)<br /><br />\[ \text{Number of sides} = \frac{360^{\circ}}{180^{\circ} - 165^{\circ}} = \frac{360^{\circ}}{15^{\circ}} = 24 \]<br /><br />So, the polygon has 24 sides.<br /><br />### 8. \(177.5^{\circ}\)<br /><br />\[ \text{Number of sides} = \frac{360^{\circ}}{180^{\circ} - 177.5^{\circ}} = \frac{360^{\circ}}{2.5^{\circ}} = 144 \]<br /><br />So, the polygon has 144 sides.<br /><br />In summary:<br />- For \(174^{\circ}\), the polygon has 60 sides.<br />- For \(156^{\circ}\), the polygon has 15 sides.<br />- For \(165^{\circ}\), the polygon has 24 sides.<br />- For \(177.5^{\circ}\), the polygon has 144 sides.