1 / 18

The Polygon Angle-Sum Theorems

Discover relationships between interior and exterior angles of polygons using angle measurements, investigation, and the Polygon Angle-Sum Theorems. Investigate quadrilaterals, triangles, pentagons, hexagons, and more to solve angle problems effectively.

rgaston
Télécharger la présentation

The Polygon Angle-Sum Theorems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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.

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

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

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

  9. 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

  10. 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.

  11. 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.

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

  13. 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.

  14. Find the missing angle measures y 100 Y = 103 Z= 70 110 z 87

  15. Find each missing angle measure. z x X = 59 W = 72 Y =49 z – 13 w y z + 10 Z= 121

  16. Find each missing angle measure. 3x 4x 2x x X =36

  17. 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.

  18. Find the sum of the measures of the angles of a decagon. A decagon has 10 sides, so n = 10.

More Related