Triangle Theorems and the Five-Point Star
Mathematical Goals: Teachers will be able to
 Construct a proof of a geometric theorem.
Pedagogical Goals:
 Identify and assess the obstacles in students’ construction of proofs and formulate
strategies for giving students entry points to proof in order to make them more
accessible.
Technological Goals:
 None
Alignment to the Common Core:
 G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles are
congruent and corresponding angles are congruent; points on a perpendicular bisector
of a line segment are exactly those equidistant from the segment’s endpoints.
 G-CO.10 Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a triangle is parallel to the third side and half
the length; the medians of a triangle meet at a point.
Mathematical Practices:
1 – Make sense of problems and persevere in solving them.
3 – Construct viable arguments and critique the reasoning of others.
7 – Look for and make use of structure.
8 – Look for and express regularity in repeated reasoning.
Length of Lesson:
90 minutes
Materials Needed:
Five-point Star handout
Overview:
Participants must first construct a basic proof and discuss the pedagogical implications of such
a task in a high school classroom, such as determining the students’ entry points or obstacles
for such a task. Participants will then be tasked with proving the Law of Sines so they can
experience the same potential obstacles that a student feels (although they will have the end
result in mind). Finally, participants must make a conjecture and prove a theorem involving the
sum of the angles of a five-point star. As an extension, they may also make conjectures
concerning the sums of the angles of six-, seven-, or higher-point stars.
The figure above should be drawn on the board. Participants must then examine the figure and
write a diagram below and write a proof demonstrating the exterior angle theorem for triangles.
Pedagogical discussion:
 How do you introduce proofs to your students?
 What is your preferred format of proofs and why? (Give pedagogical reasons.)
 What could be done to make this activity more accessible for your students? (What
launching points would you give them to where they could work with autonomy?
Consider also the possibility that the teachers may work through an example of a proof
for/with their students first. If this is the method that they choose, does this provide a
format and launching point for proofs, yet inadvertently provide the misconception that
proofs are procedural?)
Use the triangle below to prove the Law of Sines.
If necessary, the facilitator can model the use of making appropriate accommodations so the
participants can have an entry point to this proof by drawing the altitude from C to segment c.
Furthermore, if necessary, the teacher may suggest right triangle trigonometry with the triangles
formed. A complete solution is below.
sin B 
sin A 
g
a
g
b
g  b sin A
g  a sin B
a sin B  b sin A
sin A sin B

a
b
Solve for g in each of the sine equations
The Five-Point Star
Examine the diagram below, then answer the question that follows:
Determine the sum of the measures of the acute angles at vertices A, B, C, D, and E. Prove
your conjecture.
How can this activity be made more accessible to your students? How can it be made to
challenge the high flyers? Which Standards for Mathematical Practice did you use when you
did this activity?
2nd activity: Sum Star (30-40 minutes)
Required Prior Knowledge: Angle Sum Theorem for Triangles, Exterior Angle Theorem, Angle
Sum for Polygons (Pentagon Angle Sum is 540 Degrees), Linear Pair Relationships
Anticipated Approaches
Using the exterior angle theorem:
Identify exterior angle theorem locations and state all instances (all numbers signify angles
unless otherwise noted).
B+7=5
B+6=4
A+8=3
A+9=4
E + 10 = 2
E + 11 = 3
D + 12 = 1
D + 13 = 2
C + 14 = 5
C + 15 = 1
Notice that right side only consists of angles from
the interior pentagon, each being referenced twice.
Since pentagonal angle sum is 540, twice that would
be 1080 degrees. We can use this notion by
summation of all the equations, setting the entire
left side equal to 1080 degrees.
Left Side Sum has each acute star angle referenced twice, so students tend to combine:
2B + 7 + 6 + 2A + 8 + 9 + 2E + 10 + 11 + 2D + 12 + 13 + 2C + 14 + 15 = 1080°
However, a trend is noticed when the A’s, B’s, etc. are not written as combined like terms: Items
in red represent the three angles from surrounding triangles:
B + B + 7 + 6 + A + A + 8 + 9 + E + E + 10 + 11 + D + D + 12 + 13 + C + C + 14 + 15 = 1080°
Using angle sums for a triangle, 180 degrees can be substituted in for any representation of the
sum of three angles:
B + 180° + A + 180° + E + 180° + D + 180° + C + 180° = 1080°
Combining like terms (numerical angle measures) reveals:
B + A + E + D + C = 180°
(Solution)
The Angle Sum of a Triangle Method Solution:
Identify Triangles: AD5, AC2, BE1, BD3, CE4 and apply angle sum theorem to each
A + D + 5 = 180°
A + C + 2 = 180°
B + E + 1 = 180°
B + D + 3 = 180°
E + C + 4 = 180°
Once all triangles are represented, students
must notice that each letter is represented
twice. These equations can be added as a
system to achieve the following all-inclusive
equation:
2A + 2B + 2C + 2D + 2E + 1 + 2 + 3 + 4 + 5 = 900°
Apply Pentagonal Angle Sum: 1 + 2 + 3 + 4 + 5 = 540° ---> Substitute in for those angles:
2A + 2B + 2C + 2D + 2E + 540° = 900°
2A + 2B + 2C + 2D + 2E = 360°
----> Solve for Sum of A thru E.
Factor: 2(A + B + C + D + E) = 360°
Divide to reveal summation of angles A thru E:
A + B + C + D + E = 180°.
Alternate proof (in not-so-mathematical language):
Consider the quadrilaterals formed by taking a corner and the three remote corners of the inner
pentagon. Adding these angles gives 5(360°)=1800°. However, each of the angles interior to
the inner pentagon was counted three times, meaning that we added 3(540°)=1620° too much.
Therefore, the sum of the angles at the points is 180°.
Enrichment:
Participants should form conjectures about the sums of the acute angles at the vertices of six
and seven point stars and form a proof.
Number of points on the star
5
6
7
n
Sum of acute angles at the vertices
180°
360°
540°
(n-4)180°