The Polygon AngleSum Theorems Geometry 12 0 Students

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 NOTICE! 4 Sides / 1 Diagonal / 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 x 3 540º NOTICE! 5 Sides / 2 Diagonals / 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 T#1 180º T#2 T#3 180º x 4 720º NOTICE! 6 Sides / 3 Diagonals / 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 2 x 100 117 105 2 x 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 • 18 –gon Interior = 160 • 100 -gon Exterior = 30 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 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 c 3 b 180º 2 3 + c = 180º 1 + 2 + 3 + a + b + c = 540º 1 + 2 + 3 + 180º = 540º 1 + 2 + 3 = 360º The sum of the exterior angles of ANY triangle is 360º.

Graphic Sum of the Exterior Angles 1 c a 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 180º a 2 + 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 W = 72 X = 59 Y =49 z – 13 w Z= 121 y z + 10

Find each missing angle measure. 3 x 4 x 2 x 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 2 m 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.