a To determine a more appropriate name, we need to know the measure of the polygon's interior angles and number of sides.
Interior Angles
Since the polygon is regular, its interior angles must be congruent. Also, an interior angle in a polygon forms a linear pair with its exterior angle which means they are supplementary. If we label the measure of the interior angle θ, we can write and solve an equation containing θ.
θ+120^(∘)=180^(∘) ⇔ θ=60^(∘)
Each interior angle has a measure of 60^(∘).
Number of Sides
The sum of the measures of the exterior angles of any polygon is 360^(∘). Therefore, if we call the number of sides in the polygon n, we can write and solve an equation containing n.
60^(∘) n=360^(∘) ⇔ n=3
The polygon has 3 sides. This means the polygon is an equilateral triangle.
b A quadrilateral is a polygon with four sides and vertices. By substituting n=4 into the expression 180^(∘)(n-2), we can determine the sum of the interior angles.
The sum of the interior angles of a quadrilateral is 360^(∘). Since the four angles are all equal, each angle is 360^(∘)4=90^(∘). A quadrilateral with four right angles is called a rectangle.
c Like in Part B, we know that the sum of the interior angles of any polygon can be expressed as 180^(∘)(n-2), where n is the number of sides. By equating this expression with the sum of the interior angles, we get the following equation.
180^(∘)(n-2)=1260^(∘)
Let's solve this equation for n.
