The Polygon Angle-Sum Theorems
Geometry 12.0 – Students find and uses
measures of interior and exterior angles of
triangles and polygons to classify figures and
solve problems.
Geometry 13.0- Students prove relationships
between angles in polygons by using properties
of exterior angles.
Quadrilateral Investigation
• The sum of all the interior angles of a
360º
quadrilateral is _____?
– Let’s investigate!
Triangle 2
Triangle 1
Sum is
180º
Sum is
180º
NOTICE!
4 Sides / 1 Diagonal / 2 Triangles
2 Triangles = 2(180º) = 360º
180º
x 2
360º
Pentagon Investigation
• The sum of all of the interior angles of a
pentagon is _____?
540º
– Let’s investigate!
Triangle
#1
Triangle
#3
Triangle
#2
Sum is
180º
Sum is
180º
Sum is
180º
180º
x 3
540º
NOTICE!
5 Sides / 2 Diagonals / 3 Triangles
3 Triangles = 3(180º) = 540º
Hexagon Investigation
• The sum of the interior angles of a hexagon is
720º
____?
– Your turn to investigate!
180º
T#4
180º
T#1
T#2
T#3
180º
180º
180º
x 4
720º
NOTICE!
6 Sides / 3 Diagonals / 4 Triangles
4 Triangles = 4(180º) = 720º
Make a Table
Name
No. of
Sides
Diagonals
Drawn
No. of
Triangles
3
0
1
1(180º) = 180º
Quadrilateral
4
1
2
2(180º) = 360º
Pentagon
5
2
3
3(180º) = 540º
Hexagon
6
3
4
4(180º) = 720º
Triangle
Drawing
Interior Angle
Sum
Name
Drawing
No. of
Sides
Diagonals
Drawn
No. of
Triangles
Interior Angle
Sum
Heptagon
7
4
5
5(180º) = 900º
Octagon
8
5
6
6(180º) = 1080º
Nonagon
9
6
7
7(180º) = 1260º
Decagon
10
7
8
8(180º) = 1440º
25-gon
25
22
23
23(180º) = 4140º
n-gon
n
n–3
n–2
(n – 2)180º
Find the missing angle measures
X = 60
120
X = 103
100
2x
117
105
2x
x
115
x
x
Find the missing angle measures
X = 113
x
X = 145
62
116
x
x+6
120
135
129
140
151
125
130
135
Find the measures of an interior angle and
an exterior angle of each regular polygon.
• Pentagon ( 5-sides)
Interior = 108 Exterior = 72
• Dodecagon (12-sides)
Interior = 150
Exterior = 30
• 18 –gon
Interior = 160
• 100-gon
Exterior = 20
Interior = 176.4
Exterior = 3.6
Exterior Angles of Polygons
• The exterior angle of a polygon will form a
linear pair with an interior angle.
Example:
180º
Interior Exterior
Angle
Angle
Remember: Linear Pairs are Supplementary.
Sum of the Exterior Angles
• The sum of the exterior angles of a triangle is
_____.
a + b + c = 180º
Let’s
1 + a = 180º
2 + b = 180º
Investigate: 1 a
+
180º
180º
3 + c = 180º
1 + 2 + 3 + a + b + c = 540º
1 + 2 + 3 + 180º = 540º
c
3
b
180º
2
1 + 2 + 3 = 360º
The sum of the exterior angles of
ANY triangle is 360º.
Graphic Sum of the Exterior Angles
1
a
c
b
2
3
The Sum is 360º
Exterior Angles of a Quadrilateral
• The sum of the exterior angles of a
quadrilateral is _____?
a + b + c + d = 360º
180º
1
a
2
180º
+
b
1 + 2 + 3 + 4 + a + b + c + d = 720º
180º
d
4
c
180º
1 + a = 180º
2 + b = 180º
3 + c = 180º
4 + d = 180º
3
1 + 2 + 3 + 4 + 360º = 720º
1 + 2 + 3 + 4 = 360º
The sum of the exterior angles of
ANY quadrilateral is 360º.
Find the missing angle measures
y
100
Y = 103
Z= 70
110
z
87
Find each missing angle measure.
z
x
X = 59
W = 72
Y =49
z – 13
w
y
z + 10
Z= 121
Find each missing angle measure.
3x
4x
2x
x
X =36
The figure has 4 sides, so n = 4.
m < X + m < Y + m < Z + m < W = < (4 – 2)(180)
Polygon Angle-Sum Theorem
m X + m Y + 90 + 100 = 360
Substitute.
m X + m Y + 190 = 360
m X + m X = 170
2m X = 170
m X = 85
Simplify.
Substitute m X for m Y.
Simplify.
Divide each side by 2.
Find the sum of the measures of the angles of a decagon.
A decagon has 10 sides, so n = 10.