Grade 10, La Cañada High School, La Cañada, California

**Question:**

- Find the sum of the angles formed at the tips of both stars
- Find a formula for the sum of the sum of the angle measurements at the tips of an n-pointed star.

- The sum of the angles formed at the tips of the five pointed star is 180; the sum of the angles formed at the tips of the six pointed star is 360.
- The formula for the sum of the angle measurements at the tips of an n-pointed star is f(n)=180(n)-720 where n is an integer greater than 4.

Let's start off with the five pointed star:

If you get confused at any time, just refer back to the top diagram.

Before we jump into the problem, lets get the facts straight:

- Based on the diagram we can see there are five triangle whose bases make up a pentagon, which all together make up this figure.
- We also see that the polygons that make this figure are convex polygons since if I select two point inside the polygons, the segment between the two points do not exit the polygon.
- There is a theorem that states that the sum of the measures of the angles of any convex polygon, one such angle at the vertex, is 360.
- The exterior angle of a convex polygon is an angle that is formed when the side of a polygon is extended in one direction (like a ray), and is supplemmentary to an vertex angle of a polygon, and that only one line that contains a side can hold an exterior angle at a time.
- Finally, there are two sets of exteriors angles in every convex polygon.

With that out of the way, we can attact the problem.

Based on this diagram, we can see that we have:

- Pentagon FGHIJ
- Triangle AFG
- Triangle BGH
- Triangle CHI
- Triangle DIJ
- Triangle EJF

I labeled the diagram in the following way:

Red angles- Set 1 of the exterior angles of Pentagon FGHIJ

Blue Angles- et 2 of the exterior angles of Pentagon FGHIJ

Green Angles- Vertex angles

- After labeling the diagram, we can see that the measurements of the red exterior angles and the blue exterior angles each equal 360, show all together the sum of both sets of exterior angle is 720
- We can also see that the exterior angles of the pentagon makes up each set of the base angles of the triangle.

Lets deal with the five triangles now:

- The measurements of the angles that makes up the triangle equals 180
- The sum of all the angles of the 5 triangles equals 180(5), or 900
- We can setup an equation to find the the measurements (here the sum of the measurements) of the triangles:Ê 5[measurement of the vertex angle+measurement of one set of the base angles+measurement of the other set of the base angles=180],
- Next, we can distribute the 5 to get [(5)measurement of the vertex angle+(5)measurement of one set of the base angles+(5)measurement of the other set of the base angles=900]
- Since we know the sum of both sets of exterior angle (also the sum of the base angles), we can substitute 360 into the equation, we get (5)measurement of the vertex angle+720=900, or (5)measurement of the vertex angle=180

**If you are still interested, continue on:**

For the six pointed the star, we can do ALL of that work again, or we can take a shortcut...

We see that the figure is made up of two triangle, highlighted in the figure.Ê We then can say that the sum of the vertex angles is 180(2), or 360.Ê We CAN do ALL that work to arrive at the same answer, but I think I would be too time consuming.

To find the equation, we need to find the varible:

Number of triangles and the sides of the polygon that form the n-pointed figure.Even though the number of sides of the polygons changes, the sum of each set of exterior angles doesn't change ( 2 sets of 360)

We now can refer back to our original equation, and insert the constant and varible to get:

(n)measurement of the vertex angle+720=180(n) or sum of vertex angles [or (f)n] =180(n)-720.If we set up the equation to set an inequality, we get n>4 (greater than because the function needs to be greater than 0) and since we can't have numbers other than postive integers, we can see that f(n)=180(n)-720 where n is an integer greater than 4.

If you were able to stay interested, I thank you for your attention and I hope you understand.

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