1.) Essential Question How does the number of sides in convex polygons relate to the sums of the measures of he exterior and interior angles?

To understand how the number of sides in convex polygons relates to the sums of the measures of the exterior and interior angles, we can break down the problem step by step. ### Step 1: Understanding Interior Angles For a convex polygon with \( n \) sides, the sum of the interior angles can be calculated using the formula: \[ S_i = (n - 2) \times 180 \] where \( S_i \) is the sum of the interior angles. ### Step 2: Understanding Exterior Angles The sum of the exterior angles of any convex polygon, regardless of the number of sides, is always: \[ S_e = 360 \] where \( S_e \) is the sum of the exterior angles. ### Step 3: Relationship Between Interior and Exterior Angles For each interior angle \( i \) and its corresponding exterior angle \( e \), the relationship can be expressed as: \[ i + e = 180 \] This means that each pair of interior and exterior angles at a vertex adds up to \( 180 \) degrees. ### Step 4: Total Angles in Terms of Sides If we consider the total number of angles in the polygon: - There are \( n \) interior angles. - There are \( n \) exterior angles. Thus, we can express the total measures of the angles in terms of the number of sides \( n \): - Total interior angles: \( S_i = (n - 2) \times 180 \) - Total exterior angles: \( S_e = 360 \) ### Conclusion The relationship can be summarized as follows: - The sum of the interior angles increases with the number of sides, specifically by \( 180 \) degrees for each additional side. - The sum of the exterior angles remains constant at \( 360 \) degrees, regardless of the number of sides. In summary, while the sum of the interior angles of a convex polygon increases with the number of sides, the sum of the exterior angles remains constant. This illustrates a fundamental property of polygons and their angles.