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Interior Angle Sums
1. Complete the chart.
Diagram
Number
of sides
Sum of
interior angles
Understanding
The sum of the angles in any triangle is
180o.
3
180
4
5
n
2. a) Determine the sum of the interior angles in a polygon with 15 sides. Show your work.
b) Determine the number of sides in a polygon if the sum of the interior angles is 5400.
Show your work.
Interior Angle Sums (continued)
3. Derek is building a deck for his summer job in the shape of a regular octagon.
a) Define: regular octagon
?
b) Determine the measure of the interior angles of the deck.
Show your work.
4. A Canadian $1 coin, known as a loonie, is a regular polygon with 11 sides, called an
undecagon.
a) Define a regular polygon with 11 sides.
b) Determine the sum of the interior angles of the loonie.
c) What is the size of one of the interior angles?
Sum of the Exterior Angles of a Triangle (GSP®4 file)
Angles Triangle.gsp
Sum of the exterior angles of a triangle.
Sum of the exterior angles of a quadrilateral
Exterior angles of a triangle are shown.
Exterior angles of a quadrilateral are shown.
If we decrease the size of the triangle, what do you notice
about the sum of the exterior angles?
If we decrease the size of the quadrilateral,
what do you notice about the sum of
the exterior angles?
B
B
Click on the action bar below
Click on the action bar below
Make the quadrilateral smaller
A
Make the triangle smaller
C
A
Reset the quadrilateral
Reset the triangle
C
Show Measurements
D
Exterior Angle Demonstration
1. Drag each vertex and
observe the angle
measurements.
2. One less side
3. Drag the vertex of
angle 2 or 3 and
observe the sum.
4. Another side less
Angle Measurements
Angle
Angle
Angle
Angle
Angle
1
2
5
5. What do you notice
about the sum?
6.
Reset
7.
Shrink polygon
Stretch back out
8.
9. Make a conclusion.
3
4
Sketch Restriction:
The polygon must remain convex.
This sketch was based on a similar sketch from "Exploring Geometry"
1
2
3
4
5
= 72.0
= 72.0
= 72.0
= 72.0
= 72.0
Show Sum
Sum of the Interior Angles of a Polygon (GSP®4 file)
(continued)
Angles Polygon.gsp
Sum of the Interior Angles of a Polygon
Tearing Corners
1 Triangle
2
TRIANGLE -The Sum of the Interior Angles
A
Show the Sum of the Angles
These two investigations simulate tearing off the
corners of the polygon and joining them.
Reset
Quadrilateral
Diagonal Divisions
3 Summary
B
These investigations use the fact that any polygon
can be divided into triangles by drawing
diagonals. It can also be used to explore the
pattern of the sum of the angles and number of
sides.
4
Quadrilateral
5
Pentagon
6
Hexagon vs. Regular Hexagon
Tiny Triangles
7 Quadrilateral
If you move the vertices of the triangle
does the sum of the angles remain the
same?
Show SUM
This investigation uses the fact that any polygon can be
divided into triangles by connecting a center point to two
of the vertices. It also uses the idea of subtracting one
complete rotation.
Interior Angles of Polygons-Looking for Patterns
QUADRILATERAL :Sum of the Interior Angles
Hypothesis: What is the sum of the interior angles of a quadrilateral?
Show The Sum of the Angles
C
Number of
Sides
Diagram
Reset
Sum of Interior
Angles
C
B
3
180
Show me!
4
If you move the vertices of the quadrilateral does
the sum remain the same?
Show SUM
A
D
Show me!
5
K
L
M
P
N
O
Understanding
FACT:
The sum of the angles in
any triangle is 180.
The interior of a quadrilateral
has two triangles, and sum
sum of the angles in each
triangle is 180, so
2 x 180 = 360
Sum of the Interior Angles of a Polygon (GSP®4 file)
Examining the Interior Angles of a Quadrilateral
Return
Examining the Interior Angles of a Pentagon
Drag any point to form new quadrilateral.
D
C
Return
I
Drag any point to form new quadrilateral.
Number of Triangles = 3
H
J
Show Angle Measurements
G
A
B
F
Show Angle Measurements
Hexagon
Drag any vertex
K
Regular Hexagon
J
F
Drag point A
L
I
Z
B
E
The rest of the angles in the triangles form the interior
angles of the quadrilateral.
G
H
Sum of the Interior Angles of a Quadrilateral
Quadrilateral WXYZ consists of 4 triangles. The 4
triangles share vertex O inside the quadrilateral.
The sum of the angles at point O is 360 (since
there are 360 in one complete rotation).
A
D
Show Angle Measurements
Show Angle Measurements
Show Triangle Sub-Divisions
Show Triangle Sub-Divisions
Show Sum of Interior Angles
Show Sum of Interior Angles
W
C
Therefore,
Sum of the interior angles in WXYZ
= (sum of interior angles in 4 triangles) - 360 
= 4 x 180 - 360
= 360
Y
O
mXWZ = 82
mWZY = 106
mZYX = 86
mYXW = 87
Sum of Interior Angles = 360
X
Return