The Polygon Angle-Sum Theorems

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! 3 Triangles = 3(180º) = 540º 5 Sides / 2 Diagonals / 3 Triangles

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 0 Triangle 3 1 1(180º) = 180º Quadrilateral 4 1 2 2(180º) = 360º 5 3(180º) = 540º Pentagon 2 3 Hexagon 6 4(180º) = 720º 3 4

Heptagon 5 5(180º) = 900º 7 4 Octagon 8 5 6 6(180º) = 1080º Nonagon 9 7 6 7(180º) = 1260º Decagon 10 7 8(180º) = 1440º 8 25 22 23 23(180º) = 4140º 25-gon n – 2 (n – 2)180º n-gon n n – 3

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 62 116 x 120 x + 6 135 129 125 140 151 130 135

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 3 c 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

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. 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.

The Polygon Angle-Sum Theorems