Question

1. respond to the following questions showing all work. you may use the fact that the sum of all interior angles of a polygon with n sides is given by (n - 2)180°. a. define what it means for a polygon to be regular, and give an example of a regular quadrilateral. b. in a certain regular polygon, the measure of each of the interior angles is 176°. how many sides does the polygon have?

Explanation:

Step1: Define regular polygon

A regular polygon is a polygon that is equiangular (all interior angles are equal) and equilateral (all side - lengths are equal). An example of a regular quadrilateral is a square, where all four sides are of equal length and all four interior angles are 90°.

Step2: Set up equation for part b

Let \(n\) be the number of sides of the regular polygon. The sum of the interior angles of a polygon is \((n - 2)\times180^{\circ}\). Since the polygon is regular, each interior angle \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\). We are given that \(\theta = 176^{\circ}\), so we set up the equation \(\frac{(n - 2)\times180}{n}=176\).

Step3: Solve the equation

First, multiply both sides of the equation \(\frac{(n - 2)\times180}{n}=176\) by \(n\) to get \((n - 2)\times180=176n\). Expand the left - hand side: \(180n-360 = 176n\). Then subtract \(176n\) from both sides: \(180n-176n-360=0\), which simplifies to \(4n-360 = 0\). Add 360 to both sides: \(4n=360\). Divide both sides by 4: \(n = 90\).

Answer:

a. A regular polygon is an equiangular and equilateral polygon. A square is a regular quadrilateral.
b. 90