After some geometry play with doodles and a protractor, Stu Dent made this claim:
Construct a star using five connected straight lines. The sum of the angles in the five points of the star will always equal a constant value.
Is Stu correct?
If he is correct, what is the constant?
If he is incorrect, what is the range of values for this sum?
Try to find different ways to explore this problem or prove your conjectures as a check on your solution. This is a fun problem to play with using GSP or some other dynamic geometry software.
Hint: First, using either a straight-edge or GSP, draw many different star configurations as in the examples shown, being sure to vary from the standard five-pointed star whose angles are all congruent. Measure and add the five angles. (At the very least, cut out the stars and tear off the star's points and add these five angles visually.) What sum do you get?
Now, try to list the geometrical properties and theorems that might help prove this number is this constant sum.
Solution Commentary: The constant sum is 180 degrees. Many different proofs of this are possible, and my students seem to invent new ones each time I give them this problem. Try using any of the following:
Once you have found one satisfying proof, explore what happens if one of the star's angle vertices is moved to the interior of the original pentagon? Or, what happens to the angle sum if you use a star with six, seven,...n-sides?
- Exterior angle properties for the many triangles or pentagon formed
- Inscribe the star inside a circle using three of the "outermost" vertices (the remaining two angle vertices will be either interior to the circle or on the circle as well), and then apply circle/chord angle properties
- Use a pencil to implement the Turtle-Total-Trip Theorem from Logo (if you know what that is...if you don't, you might want to find out what it is!)
- Draw a line segment between two "adjacent" angle vertices and then try an "aha!" proof which involves the visual translation of the five star angles into a single triangle using the exterior angle theorem
Though this great problem is somewhat old but relatively unknown, a nice write up of explorations of the problem can be found in Alan Lipp's "The Angles of a Star" (Mathematics Teacher, September 2000, pp. 512-516).