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 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 1 + 2 + 3 + a + b + c = 540º 180º 1 + 2 + 3 + 180º = 540º c b 2 1 + 2 + 3 = 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

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 1 + 2 + 3 + 4 + a + b + c + d = 720º 180º d c 3 1 + 2 + 3 + 4 + 360º = 720º 4 1 + 2 + 3 + 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 + 90 + 100 = 360 Substitute. m X + m Y + 190 = 360 Simplify. m X + m X = 170 Substitute m X for m Y. 2m X = 170 Simplify. m X = 85 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.