\begin{tabular}{l} Question 4 \\ The size of each interior angle of a regular polygon is \( 165^{\circ} \). How many sides does the polygon have? \\ \( \begin{array}{l}24 \\ 22 \\ 20 \\ 26\end{array} \) \\ \hline\end{tabular}

To find the number of sides in a regular polygon where each interior angle is \( 165^\circ \), you can use the formula for the interior angle of a regular polygon given by: \[ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \] Setting this equal to \( 165^\circ \): \[ \frac{(n-2) \times 180}{n} = 165 \] Cross-multiplying gives: \[ (n-2) \times 180 = 165n \] Expanding and rearranging the equation results in: \[ 180n - 360 = 165n \] \[ 15n = 360 \] \[ n = 24 \] So, the polygon has 24 sides. --- The world of polygons is fascinating! Regular polygons, where all sides and angles are equal, have been studied for centuries. They create stunning tessellations and are fundamental in various designs, from architecture to art. For example, the iconic honeycomb structure made by bees is a hexagonal tiling, showcasing nature’s preference for efficient shapes! If you're tackling similar problems, a common mistake is to misidentify the relationships between the interior and exterior angles. Remember, exterior angles always sum up to \( 360^\circ \) for any polygon. The equation for an interior angle can sometimes be tricky, but once you grasp how they relate, you'll be counting sides with ease!