Geometry

Question Analysis

The main focus of this question is two - fold. First, it aims to explore the relationship between the number of sides of a regular polygon and the measure of its exterior angle. Second, it requires calculating the measure of one interior angle of a regular decagon. This tests the understanding of fundamental properties of regular polygons.

 

Key Concept Explanation

Relationship between the number of sides () and the measure of an exterior angle () of a regular polygon: The sum of the exterior angles of any polygon is always . In a regular polygon, since all exterior angles are equal, the measure of one exterior angle . As increases, the measure of each exterior angle decreases.

Measure of an interior angle () of a regular polygon: The formula for the measure of an interior angle of a regular polygon with sides is . This formula is derived from the fact that the sum of interior angles of a polygon is , and in a regular polygon, all interior angles are congruent.

 

Step - by - Step Solution

1. Relationship between the number of sides and the measure of an exterior angle:

Given that the sum of exterior angles of any polygon is . For a regular polygon with sides, if we let the measure of each exterior angle be , then . For example, in a regular triangle (), , and in a square (), . As gets larger, the value of gets smaller.

2. Measure of one interior angle of a regular decagon:

For a decagon, .

Substitute into the formula .

First, calculate

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