Question

extension 1. a. what type of polygon is formed by drawing segments connecting the non - point vertices of an n - pointed star as in the example at the left? b. what is the measure of an interior angle of this polygon? c. what type of triangle is formed at each point? d. what is the measure of each of the base angles in the triangle? e. what is the sum of the four angles formed at each non - point vertex of the star? f. use your results from parts b, d, and e to show that the equation you wrote in exercise 2b is true for all stars.

Explanation:

Step1: Identify the polygon in part a

The polygon formed by connecting non - point vertices of an $N$-pointed star is a regular $N$-gon.

Step2: Calculate interior angle in part b

The formula for the measure of an interior angle $\theta$ of a regular $n$-gon is $\theta=\frac{(n - 2)\times180^{\circ}}{n}$. Here $n = N$, so $\theta=\frac{(N - 2)\times180^{\circ}}{N}$.

Step3: Determine triangle type in part c

The triangles formed at each point of the star are isosceles triangles.

Step4: Find base - angle measure in part d

Let the central angle of the star be $\frac{360^{\circ}}{N}$. The vertex angle of each isosceles triangle at the point of the star is related to the central angle. The measure of the vertex angle of the isosceles triangle at each point of the star is $\frac{180^{\circ}(N - 4)}{N}$. Using the angle - sum property of a triangle ($180^{\circ}$ for the sum of interior angles of a triangle), if the vertex angle of the isosceles triangle is $v=\frac{180^{\circ}(N - 4)}{N}$, and the base angles are equal, say $\alpha$, then $2\alpha+v = 180^{\circ}$. Solving for $\alpha$ gives $\alpha=\frac{180^{\circ}-\frac{180^{\circ}(N - 4)}{N}}{2}=\frac{360^{\circ}}{2N}=\frac{180^{\circ}}{N}$.

Step5: Calculate angle - sum in part e

At each non - point vertex of the star, the sum of the four angles is $360^{\circ}$ since the angles around a point sum to $360^{\circ}$.

Step6: Prove the equation in part f

The details of the equation in Exercise 2b are not given, but in general, using the above results: The interior angle of the $N$-gon from part b, the base - angle of the isosceles triangle from part d, and the angle - sum at non - point vertices from part e, we can show relationships based on the geometric properties of the star. For example, if we consider the angle - relationships around the vertices of the star and the polygon formed inside, we can use angle - sum properties of polygons and triangles to prove the relevant equation.

Answer:

a. Regular $N$-gon
b. $\frac{(N - 2)\times180^{\circ}}{N}$
c. Isosceles triangle
d. $\frac{180^{\circ}}{N}$
e. $360^{\circ}$
f. (Proof depends on the equation from Exercise 2b, but in general, use angle - sum properties of polygons and triangles as described above)