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Polygon Angle Sum Theorems
Geometry 3-4
Vocabulary
Vocabulary
Vocabulary
• Name a polygon with consecutive
vertices
Polygons
• Diagonal – A line segment that connects two
non-consecutive vertices
• Convex – No diagonal goes outside the polygon
• Concave – At least one diagonal is outside the
polygon
Vocabulary
• Equilateral – A polygon with all sides congruent
• Equiangular – A polygon with all angles
congruent
• Regular Polygon – A polygon that is both
equilateral and equiangular
Vocabulary
Equilateral
Triangle
60°
Angle Exploration
Square
90°
Angle Exploration
Regular
Pentagon
108°
Angle Exploration
Regular
Hexagon
120°
Angle Exploration
Regular
Heptagon
129°
Angle Exploration
Regular
Octagon
135°
Angle Exploration
Regular
Nonagon
140°
Angle Exploration
Regular
Decagon
144°
Angle Exploration
Regular
Dodecagon
147°
Angle Exploration
• Summarize our
results
Is there a
correlation
between the
number of sides
and the degrees?
Angle
Exploration
Sides
3
4
5
6
7
8
9
10
11
Degrees
60°
90°
108°
120°
129°
135°
140°
144°
147°
• What is the sum of the angles
in each of the polygons
Angle Exploration
• Summarize
our results
Total Degrees in
each polygon.
Do you see a
pattern?
Angle
Exploration
Sides
3
4
5
6
7
8
9
10
11
Total
180°
360°
540°
720°
900°
1080°
1260°
1440°
1620°
Degrees
60°
90°
108°
120°
129°
135°
140°
144°
147°
• The sum of the angles of the
interior angles of a convex ngon is (n-2)•180°
Polygon Angle Sum Theorem
• What are the angle measures
if the polygon is regular
• (n-2)•180°/n
Interior Angles in a Polygon
X=85
X=97
8
36
71
19.5
60
17
104
X=96
X=117
Polygon Exterior Angle Sum
Geometry 3-4b
2
1
Vertical Pair
Review – Name the angle
relationship
160°
20°
Supplementary Angles
Review – Name the angle
relationship
2
1
Linear Pair
Review – Name the angle
relationship
35°
55°
Complementary Angles
Review – Name the angle
relationship
2
1
Vertical Pair
Review – Name the angle
relationship
60°
30°
Complementary Angles
Review – Name the angle
relationship
1
2
Linear Pair
Review – Name the angle
relationship
90°
90°
Supplementary Angles
Review – Name the angle
relationship
Any of the six
angles inside
the polygon
What is an interior angle?
When the sides
of the polygon
are extended,
the angle that is
adjacent to the
interior angle
What is an exterior angle?
When the sides
of the polygon
are extended,
the angle that is
adjacent to the
interior angle
What is an exterior angle?
Spot the exterior angle
Which side do
you extend to
create an
exterior angle?
Exterior Angles
Either one, these
two angles are
congruent,
because they
are vertical
angles.
Exterior Angles
What is the
value of all the
exterior angles
in any polygon?
Exterior angle
• Get your supplies
– Printer Paper
– Straightedge
– Protractor
Exterior Angle Investigation
• Draw a large polygon on your paper
Pentagon or larger
• Extend each of its sides to form exterior
angles
Exterior Angle Investigation
• Measure all of the interior angles except one.
Write the value of the interior angle on the
angle.
• Use the polygon sum conjecture to calculate
the measure of the last angle before you
measure it.
Exterior Angle Investigation
• Use the linear pair conjecture to calculate
the value of each exterior angle.
• Write down the value on the angle
• Add up all the exterior measures
Exterior Angle Investigation
• The sum of the measures of the
exterior angles of a convex polygon,
one angle at each vertex, is 360°
Polygon Exterior Angle Sum Theorem
What is the
measure of an
exterior angle in
a regular
polygon?
Exterior angle
• The measure of each exterior angle of a
regular n-gon is 360°/n
Exterior Angle Theorem
• You can find the measure of each
interior angle of an equiangular n-gon
by using either of these formulas:
180° - 360°/n
(180°(n-2))/n
Equiangular Polygon Conjecture
• Pages 147 – 150
• 1 – 4, 12, 13, 17, 20, 47 –
49, 54, 71, 73 - 79
Homework