The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum TheoremsGeometry 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

quadrilateral is _____?– Let’s investigate!

Triangle 1

Triangle 2 Sum is180º

Sum is180º

2 Triangles = 2(180º) = 360º 4 Sides / 1 Diagonal / 2 Triangles

NOTICE!

180ºx 2360º

360º

Pentagon Investigation• The sum of all of the interior angles of a

pentagon is _____?– Let’s investigate!

Triangle#1

Triangle#2

Triangle#3

Sum is180º

Sum is180º

Sum is180º

3 Triangles = 3(180º) = 540º

NOTICE!

5 Sides / 2 Diagonals / 3 Triangles

180ºx 3

540º

540º

Hexagon Investigation• The sum of the interior angles of a hexagon is

____?– Your turn to investigate!

T#1T#2

T#3

T#4180º

180º 180º

180º

NOTICE!

6 Sides / 3 Diagonals / 4 Triangles 4 Triangles = 4(180º) = 720º

180ºx 4

720º

720º

Make a TableName Drawing No. of

SidesDiagonals

DrawnNo. of

TrianglesInterior Angle

Sum

3 0 1 1(180º) = 180º

4 1 2 2(180º) = 360º

5 2 3 3(180º) = 540º

6 3 4 4(180º) = 720º

Triangle

Quadrilateral

Pentagon

Hexagon

Name Drawing No. of Sides

Diagonals Drawn

No. of Triangles

Interior Angle Sum

7 4 5 5(180º) = 900º

8 5 6 6(180º) = 1080º

9 6 7 7(180º) = 1260º

10 7 8 8(180º) = 1440º

25 22 23 23(180º) = 4140º

n n – 3 n – 2 (n – 2)180º

Heptagon

Octagon

Nonagon

Decagon

25-gon

n-gon

Find the missing angle measures

100

x117

105 115 x x

2x 2x

X = 103X = 103X = 60 X = 60 120120

Find the missing angle measures

x

x + 6

140 151

62

135

x

129

130 135

116

120

125

X = 113X = 113 X = 145X = 145

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 = 176.4Exterior = 3.6

Exterior Angles of Polygons

• The exterior angle of a polygon will form a linear pair with an interior angle.

Example:

Interior Angle

Exterior Angle

Remember: Linear Pairs are Supplementary.

180º

• The sum of the exterior angles of a triangle is _____.

Let’sInvestigate:

Sum of the Exterior Angles

a1

The sum of the exterior angles ofANY triangle is 360º.

bc

3

2

a + b + c = 180º

1 + a = 180º2 + b = 180º3 + c = 180º+

1 + 2 + 3 + a + b + c = 540º

1 + 2 + 3 + 180º = 540º

1 + 2 + 3 = 360º

180º

180º

180º

Graphic Sum of the Exterior Angles

a1

bc

3

2

The Sum is 360º

• The sum of the exterior angles of a quadrilateral is _____?

Exterior Angles of a Quadrilateral

d c

b

a1

4

3

2

a + b + c + d = 360º

1 + a = 180º2 + b = 180º3 + c = 180º4 + d = 180º+

1 + 2 + 3 + 4 + a + b + c + d = 720º

1 + 2 + 3 + 4 + 360º = 720º

1 + 2 + 3 + 4 = 360º

The sum of the exterior angles ofANY quadrilateral is 360º.

180º

180º

180º

180º

y

110 87z

100

Find the missing angle measures

Z= 70

Y = 103

Find each missing angle measure.

x

w yz – 13

z + 10

z

Z= 121

X = 59

Y =49

W = 72

Find each missing angle measure.

2x

x

4x

3x

X =36

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 Simplify.

2m X = 170 Simplify. m X = 85 Divide each side by 2.

m X + m X = 170 Substitute m X for m Y.

The figure has 4 sides, so n = 4.

Find the sum of the measures of the angles of a decagon.

A decagon has 10 sides, so n = 10.