DAY 2 - SUM OF INTERIOR ANGLES IN A TRIANGLE
POLYGON CONSTRUCTION
Title
Goal
Triangulation
Investigate the sum of interior angles in a triangle, and apply
to determine the sum of interior angles in a polygon.
Standard
Addressed
Current Standards: 5MG2.1; 5.MG2.2; 6MG2.2; 6.MG 2.3
Materials for
Teacher
Materials for
Students
Description
Reflection
Looking Ahead
Common Core Standards: 4.MD.7; 5.G.CA-1; 5.G.CA-2;
7.G.2
Overhead, handouts, colored pencils, Geoboards, rubber
bands, compass, glue stick, construction paper, fraction circles
Lesson handouts, colored pencils, Geoboards, rubber bands
compass, glue stick, construction paper, fraction circles
We look at the sum of the interior angles of a triangle using
manipulatives. Next, we apply the learning to determine the
sum of the interior angles in a polygon. We use this formula
to determine the measure of each angle in a regular polygon.
Triangles are the simplest polygons. Many properties of
polygons can be derived by “triangulating” them.
Link to text
In the previous lesson we talked about straight angles (an angle whose measurement is 180° or
two right angles). In this lesson we investigate the sum of the interior angles in a triangle.
mABC = 180°
C
B
A
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Sum of Interior Angles in a Triangle
Thales (ca 624BC – 547BC) was the first one to notice properties of triangles in the tiles of
Egyptian temples. He noticed that if three of the tiles, which were in the shape of equilateral
triangles, were lined up their angles formed a line (or a straight angle).
We use Thales’ idea to conjecture the sum of the interior angles in a triangle. Cut a row of three
triangles from Handout 1. Glue the strip on construction paper. Label the interior angles as A, B
and C in each triangle. Next, cut each triangle. Is it possible to line up the tiles just like Thales
did? Glue them below. What can you say about the sum of the interior angles in your triangle?
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Of course, Thales started thinking if he could do the same in other cases. Let’s construct three
triangles with Anglegs. Use one red Angleg, one blue Angleg and one yellow Angleg to
construct each triangle. Try to line the angles up. Your angles are blue and yellow, red and
yellow, and blue and red. Do your angles line up to a straight angle?
Since Thales couldn’t line up all possible tiles, he had to work with other ideas. We will do the
same. Use a ruler to construct a triangle on construction paper. Label the angles as B, R and Y.
Use different colors to color each angle. Cut the triangle. Cut the angles and try to line them
like Thales lined up the tiles. Is it possible? What does that tell you about the sum of the interior
angles in your triangle? Glue the lined up angles below…. And if possible the triangle (or
what’s left of it!)
Before we continue, we need to clarify that the activity above DOES NOT prove that the sum of
the interior angles in a triangle is equal to two right angles or 180°. We have only shown this to
be the case in specific instances and they do not cover ALL the possible triangles that exist. The
proof of this result will come later when we meet Euclid and parallel lines.
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Sum of Interior Angles in a Quadrilateral
Thales also noticed that the sum of interior angles in a square was predictable. What is the sum
in this case?
We will use a process called triangulation to determine the sum of the interior angles in
quadrilateral ABCD. Construct a point P inside the quadrilateral (P is called an interior point).
Construct the segments PA, PB, PC and PD. How many triangles are formed? Label the angles
of these triangles as {A1, A2, B1, B2, C1, C2, D1, D2, P1, P2, P3, P4} in a counterclockwise
manner.
D
C
A
B
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Since we have four triangles and the sum of the interior angles in each is 180° the sum of all of
the interior angles is 4 * 180°. However, we do not want the sum of the angles around P. This
sum is 360°. What is the sum of the interior angles in a quadrilateral?
Sum of the Interior Angles in a Pentagon
Proceed as we did above to determine the sum of the interior angles in a pentagon. Use the
figure below. Clearly write each step.
D
E
C
A
B
Sum of Interior Angles in a Decagon (ten sided polygon)
Proceed as above to determine the sum of the interior angles in a decagon:
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Sum of the Interior Angles in a Pentadecagon (fifteen sided polygon)
Proceed as we did with the previous polygons to determine the sum of the interior angles in a
pentadecagon.
D1
D2
D
C2
E
C1
C
E1
B2
E2
B1
A
B
A1
A2
The power of mathematics relies on the use of symbols and variables.
Imagine that we have a polygon with N sides. We will proceed as above but with no figure.
Imagine that we pick up an interior point P.
How many triangles can we form using P as a vertex of the triangles?
What is the sum of the interior angles of these triangles?
Are there any angles that we do not need?
WRITE THE SUM OF INTERIOR ANGLES IN A POLYGON HERE:
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REGULAR POLYGONS
In a regular polygon all interior angles and sides have the same measure (they are called
congruent). First, we will use fraction circles to construct some regular polygons. Next, we use
the formula for the sum of the interior angles in a polygon to determine the measure of each
interior angle.
CONSTRUCTION OF A REGULAR PENTAGON
The Measure of Each Interior Angle in Regular Pentagon
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Handout 1
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