Question

1. choose the correct answer.

the polygon interior angle sum theorem states that the sum of the measures of the interior angles of a convex polygon with n sides is ____.
\\( \frac { n ^ { 2 } - 3 n } { 2 } \\)
\\( ( n - 2 ) 180 ^ { \circ } \\)
triangle sum theorem
\\( 180 ^ { \circ } \\)

1. choose the correct answer.

the polygon exterior angle sum theorem states that the sum of the measures of the exterior angles, one at each vertex, of a convex polygon is ____.
\\( 360 ^ { \circ } \\)
triangle sum theorem
\\( ( n - 2 ) 180 ^ { \circ } \\)
\\( 180 ^ { \circ } \\)

Explanation:

13)

The Polygon Interior Angle Sum Theorem is a well - known geometric formula. For a convex polygon with \(n\) sides, we can divide the polygon into \((n - 2)\) non - overlapping triangles. Since the sum of the interior angles of a triangle is \(180^{\circ}\), the sum of the interior angles of the polygon is \((n-2)\times180^{\circ}\). The formula \(\frac{n^{2}-3n}{2}\) is not related to the interior angle sum of a polygon. The Triangle Sum Theorem is about the sum of angles in a triangle (\(180^{\circ}\)), not a general polygon.

The Polygon Exterior Angle Sum Theorem: If we consider a convex polygon, when we "walk" around the polygon, the total turn (sum of exterior angles) is equivalent to a full rotation. A full rotation is \(360^{\circ}\). The formula \((n - 2)180^{\circ}\) is for the sum of interior angles. The Triangle Sum Theorem (\(180^{\circ}\)) is about triangles.

Answer:

\((n - 2)180^{\circ}\)

14)