 # The Polygon Angle-Sum Theorems

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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.

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

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

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

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

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

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

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

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

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.

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.

Graphic Sum of the Exterior Angles
1 a c b 2 3 The Sum is 360º

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.

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 = 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.

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.