For exercises \( 1-4 \), find the sum of the interior angles and the measure of each interior angle for the given regular polygons. Round to the nearest hundredth as needed. \( \begin{array}{ll}\text { 1. } 12 \text {-gon } & \text { 2.102-gon } \\ \text { 3. } 90 \text {-gon } & \text { 4. } 36 \text {-gon }\end{array} \)

To find the sum of the interior angles and the measure of each interior angle for regular polygons, we can use the following formulas: 1. The sum of the interior angles \( S \) of a polygon with \( n \) sides is given by: \[ S = (n - 2) \times 180 \] 2. The measure of each interior angle \( A \) of a regular polygon is given by: \[ A = \frac{S}{n} = \frac{(n - 2) \times 180}{n} \] Now, let's calculate these values for each of the given polygons. ### 1. For the 12-gon: - \( n = 12 \) - Sum of interior angles: \[ S = (12 - 2) \times 180 = 10 \times 180 = 1800 \] - Measure of each interior angle: \[ A = \frac{1800}{12} = 150 \] ### 2. For the 102-gon: - \( n = 102 \) - Sum of interior angles: \[ S = (102 - 2) \times 180 = 100 \times 180 = 18000 \] - Measure of each interior angle: \[ A = \frac{18000}{102} \approx 176.47 \] ### 3. For the 90-gon: - \( n = 90 \) - Sum of interior angles: \[ S = (90 - 2) \times 180 = 88 \times 180 = 15840 \] - Measure of each interior angle: \[ A = \frac{15840}{90} \approx 176 \] ### 4. For the 36-gon: - \( n = 36 \) - Sum of interior angles: \[ S = (36 - 2) \times 180 = 34 \times 180 = 6120 \] - Measure of each interior angle: \[ A = \frac{6120}{36} = 170 \] Now, let's summarize the results: 1. **12-gon**: - Sum of interior angles: \( 1800 \) - Measure of each interior angle: \( 150 \) 2. **102-gon**: - Sum of interior angles: \( 18000 \) - Measure of each interior angle: \( 176.47 \) 3. **90-gon**: - Sum of interior angles: \( 15840 \) - Measure of each interior angle: \( 176 \) 4. **36-gon**: - Sum of interior angles: \( 6120 \) - Measure of each interior angle: \( 170 \) These calculations provide the required sums and measures for each polygon.