Final answer:
The missing values for a pentagon are 540° for the sum of the interior angles, 108° for the average measure of an interior angle, and 72° for the average measure of an exterior angle. The sum of the measures of the exterior angles is always 360°, regardless of the number of sides in a polygon.
Step-by-step explanation:
To find patterns within the table of data and extend the patterns to complete the rest of the table, first, observe the given values. For any polygon, the sum of the interior angles can be calculated using the formula Sum of the Interior Angles = (n - 2) × 180°, where n represents the number of sides. Using this formula, we can fill in the missing value for the pentagon (5-sided polygon): (5 - 2) × 180° = 540°.
The Average Measure of an Interior Angle can be found by dividing the sum of interior angles by the number of sides. For the pentagon: 540° ÷ 5 = 108°.
The Average Measure of an Exterior Angle is the supplementary angle of the interior angle, since interior and exterior angles sum up to 180°. Therefore, for the pentagon: 180° - 108° = 72°.
For column 5, a crucial observation is that the Sum of the Measures of the Exterior Angles for any polygon is always 360°, no matter the number of sides. This is because the exterior angles, when extended, form a circle around the polygon. This consistent sum remains true whether the polygon is a triangle, square, or any other n-sided polygon.
Correcting the data for the pentagon and the hexagon:
- Pentagon: Sum of the Interior Angles = 540°, Average Measure of an Interior Angle = 108°, Average Measure of an Exterior Angle = 72°, Sum of the Measures of the Exterior Angles = 360°
- Hexagon: Average Measure of an Exterior Angle = 60° (since 180° - 120° = 60°), Sum of the Measures of the Exterior Angles = 360°