1. 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 Students prove relationships between angles in polygons by using properties of exterior angles.

2. Quadrilateral Investigation
The sum of all the interior angles of a quadrilateral is _____?
Let’s investigate!
360º
Triangle 2
Sum is
180º
180º
x 2
Sum is
180º
Triangle 1
360º
NOTICE!
4 Sides / 1 Diagonal / 2 Triangles
2 Triangles = 2(180º) = 360º

3. Pentagon Investigation
The sum of all of the interior angles of a pentagon is _____?
Let’s investigate!
540º
Sum is
180º
Sum is
180º
180º
Triangle #3
Triangle #1
x 3
Triangle #2
Sum is
180º
540º
NOTICE!
5 Sides / 2 Diagonals / 3 Triangles
3 Triangles = 3(180º) = 540º

4. Hexagon Investigation
The sum of the interior angles of a hexagon is ____?
Your turn to investigate!
720º
180º
180º
T#4
180º
x 4
T#1
T#3
T#2
720º
180º
180º
NOTICE!
6 Sides / 3 Diagonals / 4 Triangles
4 Triangles = 4(180º) = 720º

5. Make a Table Name Drawing No. of Sides Diagonals Drawn
No. of Triangles
Interior Angle Sum
Triangle 3 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

6. No. of Sides Diagonals Drawn No. of Triangles 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

7. Find the missing angle measures
X = 60
120
X = 103
100 2x 2x
117 x
105
115 x x

8. Find the missing angle measures
X = 113
X = 145 x
x + 6
140
151 62
135
116 x
120
129
125
130
135

9. Pentagon ( 5-sides) Dodecagon (12-sides) 18 –gon 100-gon
Find the measures of an interior angle and an exterior angle of each regular polygon.
Pentagon ( 5-sides)
Dodecagon (12-sides)
18 –gon
100-gon
Interior = 108
Exterior = 72
Interior = 150
Exterior = 30
Interior = 160
Exterior = 20
Interior = 176.4
Exterior = 3.6

10. Exterior Angles of Polygons
The exterior angle of a polygon will form a linear pair with an interior angle.
Example:
180º
Interior
Angle
Exterior
Angle
Remember: Linear Pairs are Supplementary.

11. Sum of the Exterior Angles
The sum of the exterior angles of a triangle is _____.
Let’s
Investigate:
a + b + c = 180º
1 + a = 180º 1
2 + b = 180º a +
3 + c = 180º
180º
a + b + c = 540º
180º
º = 540º c b 2
= 360º
180º 3
The sum of the exterior angles of
ANY triangle is 360º.

12. Graphic Sum of the Exterior Angles1 a c b 2 3
The Sum is 360º

13. Exterior Angles of a Quadrilateral
The sum of the exterior angles of a quadrilateral is _____?
a + b + c + d = 360º
180º
1 + a = 180º 1
2 + b = 180º a 2
3 + c = 180º
180º +
4 + d = 180º b
a + b + c + d = 720º
180º d c 3
º = 720º 4
= 360º
180º
The sum of the exterior angles of
ANY quadrilateral is 360º.

14. Find the missing angle measuresy
100
Y = 103
Z= 70
110 z 87

15. Find each missing angle measure.z x
X = 59
W = 72
Y =49
z – 13 w y
z + 10
Z= 121

16. Find each missing angle measure.3x 4x 2x x
X =36

17. 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 = Substitute.
m X + m Y = Simplify.
m X + m X = Substitute m X for m Y.
2m X = Simplify.
m X = Divide each side by 2.

18. A decagon has 10 sides, so n = 10.
Find the sum of the measures of the angles of a decagon.
A decagon has 10 sides, so n = 10.