Download Mathematical Discussion about Geometry

Jennifer Eldred
(Buchholz)
09/30/2014
Reflection on Mathematical Discussion about Geometry
Draw a 5-pointed star (see the attached GSP file). Label the "points" of the star A-E. Find the sum of the
angles associated with the points (measure of angle A + measure of angle B + . . .). Justify your
reasoning.
The first thing I did when I started working on this problem was to draw a few stars. I then
thought about what could possibly make up a star so that we can find some information about the
angles or each tip of the star. I saw that the shape in the middle of the star was a pentagon so initially I
assumed it was a regular pentagon. Then I figured that there were 5 triangles each sharing one edge
with the pentagon to make up a five pointed star like we used to draw in middle school all the time. I
also assumed that this five pointed star was made up of exactly 5 straight lines (I knew this to be true
because I had drawn so many in middle school).
Because the pentagon is regular that means all the interior angles are equal. The part I wasn't so
confident in was what those interior angles are supposed to add to. Now my theory was that the
exterior angles of a regular polygon should add to 360. Therefore each interior angle in the pentagon
equal to 108 degrees. It turns out this was very much a shot in the dark and I was not sure what
theorem gave the interior angles of any polygon. At very first my thought was that it was just 180
divided by the number of sides because that is how you find each angle in an equilateral triangle. But
then I got 180/5 which equals 36. Initially I labeled my pentagon with 36 as the measurement for each of
the interior angles but I knew by looking at the pentagon that each of its interior angles must be obtuse
so 36 did not make any sense. So I thought about a square which I know has equal interior angles with
measures of 90. In that case my theory of 180 divided by the number of sides was shot because 180/4 =
45 and I know it is supposed to be 90. Therefore I guessed that the exterior angles of a regular polygon
should add to 360 so that I would get obtuse interior angles for my pentagon. That is how I came up
with each interior angle of the pentagon having a measure of 108 degrees.
Since the star is drawn with 5 straight lines, we have many straight angles comprised of one
interior pentagon angle and one triangle angle. Because of this, we know each triangle is isosceles, best
seen by the diagram at the end of this paper labeled “Initial Thoughts”. Using the fact that a straight
angle always has a measure of 180 degrees, I had already decided that the interior angle of the
pentagon has a measure of 108 degrees. Therefore an angle of the triangle has to have a measure of
180-108 = 72 degrees. This logic holds for every angle that is sharing an edge with the pentagon. Now
we have proved that our triangles coming off of the pentagon are isosceles because each angle that
shares an edge with the pentagon has a measure of 72 degrees. To find the last angle in the triangle(any
triangle will do A, B, C, D, or E because they are all congruent, because two of their three angles have
shown to be the same, therefore the third must be also). Using the theorem that the sum of all three
angles of a triangle must add up to 180 degrees, we find that the tip of each triangle (angle A, angle B,
angle C, angle D and angle E) must have a measure of 36 degrees. Therefore angle A + angle B + angle C
+ angle D + angle E = 36*5 = 180 degrees.
Janae commented on Kim’s post pretty early on wondering if we could assume that the
pentagon in the center was regular or not. I thought that it did bring up an interesting distinction,
whether or not the star includes a regular pentagon. My explanation thus far obviously depends on that
fact. I did gain confidence in my answer of 180 degrees as the sum of the angles A, B, C, D, and E
because Kim arrived at that same answer as did Casey. I also thought it was very interesting that they
went the route of inscribing the star in a circle and then using sectors of circles to solve for the angles.
So then I proceeded to quickly draw a few stars and I realized that none of my stars would fit perfectly in
a circle and they looked quite lopsided without a regular pentagon inside. So then I began wondering if
that assumption was indeed valid. I was thankful that my peers cleared up my logic about the interior
angle of a regular pentagon. Even though by this point I had determined that we could not use that
assumption I was still curious if I was remembering that theorem correctly. Janae responded to my post
saying “the interior angles of a regular pentagon add to 540 degrees, so the 108 degrees per interior
angles is correct” and I had a couple other people also agree that they remembered that theorem from
geometry. So now I had the utmost confidence in my solution given a regular pentagon in the center of
the star. But the question became will that hold true for “lopsided” stars with irregular pentagons in the
center?
In reading Dr. Kinzel’s email clarifying the assumptions that we could make and in reading my
peers posts I finally saw that the way to approach this problem generically was with triangles and not
with pentagons at all. We discussed this partially in class and then it was Casey’s post, and picture really,
that made the light bulb go off for me. Below is the picture of her rationale:
You can see that the red triangle is made up of angle A and
angle C and an unknown angle. That unknown angle is
supplementary to the angle shown with a red arc in the
picture therefore the red arc angle and the unknown angle
must add to 180 degrees. Therefore the red arc angle must
be the measure of angle A plus the measure of angle C.
Similarly, the beige triangle is made up of angle B and angle
E and an unknown angle. Again the unknown angle is
supplementary to the angle shown with the beige arc in the
picture, therefore the unknown angle and the beige arc
angle must add up to 180 degrees. Therefore the beige arc
angle must be the measure of angle B plus the measure of
angle E.
Now let’s take a look at that triangle in the upper right hand of the picture, made up of the
beige arc angle, the red arc angle and angle D. By the theorem that the sum of the three angles in a
triangle must add to 180, we know that the beige arc angle plus the red arc angle plus angle D must
equal 180 degrees. Which means that:
Beige arc angle + red arc angle + angle D = 180 degrees
(angle B + angle E) + (angle A + angle C) + angle D = 180 degrees.
Therefore my initial answer of 180 degrees was correct, which makes sense because the case
where the pentagon in the interior of the star is regular is just a special case of this property. We have
now proven that given a five pointed star made up of 5 line segments and the tips of the star labeled A.
B, C, D, and E. That angle A + angle B + angle C + angle D + angle E will always equal 180 degrees.
“Initial Thoughts”
by: Jennifer Eldred